1. Introduction

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). An edge-covering of finite and simple graph \(G\) is a family of subgraphs \(H_1, H_2, \dots,H_t\) such that each edge of \(E(G)\) belongs to at least one of the subgraphs \(H_i\), \(i=1, 2, \dots, t\). In this case we say that \(G\) admits an \((H_1, H_2, \dots, H_t)\)-(edge) covering. If every subgraph \(H_i\) is isomorphic to a given graph \(H\), then the graph \(G\) admits an \(H\)-covering. A graph \(G\) admitting an \(H\)-covering is called \((a,d)\)-\(H\)-antimagic if there exists a total labeling \(f:V(G)\cup E(G) \to \{1,2,\dots, |V(G)|+|E(G)| \}\) such that for each subgraph \(H'\) of \(G\) isomorphic to \(H\), the \(H'\)-weights, $$wt_f(H')= \sum\limits_{v\in V(H')} f(v) + \sum\limits_{e\in E(H')} f(e),$$ constitute an~arithmetic progression \(a, a+d, a+2d,\dots , a+(t -1)d\), where \(a>0\) and \(d\ge 0\) are two integers and \(t\) is the number of all subgraphs of \(G\) isomorphic to \(H\).

The (super) \(H\)-magic graph was first introduced by Gutiérrez and Lladó in [1]. The \((a,d)\)-\(H\)-antimagic labeling was introduced by Inayah et al. [2].

In [3] Bača et al. investigated the super tree-antimagic total labelings of disjoint union of graphs. Bača et al. [4] showed the constructions for \(H\)-antimagicness of Cartesian product of graphs. In [5], authors proved the \(C_n\)-antimagicness of Fan graph for several difference depending on the length of the cycle. In [6, 7, 8] Umar et al. proved the existence of super \((a,1)\)-Tree-antimagicness of Sun graphs, super \((a,d)\)-\(C_n\)-antimagicness of Windmill graphs for several differences and super \((a,d)\)-\(C_4\)-antimagicness of Book graph and their disjoint union.

In this paper, we study the existence of super \((a,d)\)-\(C_3\)-antimagic labeling of a special type of a corona graph.

2. Super Cycle-antimagic labeling of Corona graph

The join of two graphs \(H_1\) and \(H_2\), denoted by \(H_1+H_2\), is the graph where \(V(H_1) \cap V(H_2)= \emptyset\) and each vertex of \(H_1\) is adjacent to all vertices of \(H_2\) [9]. When \(H_1=K_1\), this is the corona graph \(K_1 \odot H_2\). In this paper, we consider a special type of a corona graph.

Let \(K_1\) be a complete graph and \(S_n\) be a star on \(n+1\) vertices. We consider the corona graph \(G= K_1 \odot S_n\), where $$V(G):=\{v_1,v_2,x_1,x_2,\dots,x_n\}$$ and $$E(G):=\{v_1v_2,v_1x_1,v_1x_2,\dots,v_1x_n,v_2x_1,v_2x_2,\dots,v_2x_n\}$$ The corona graph \(G\) is covered by the cycles \(C_3^{(i)}\), \( 1\leq i \leq n\) and the \(C_3^{(i)}\)-weights under a labeling \(h\) is:

2.1. \(C_3\)-Supermagic labeling

Theorem 2.1. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a \(C_3\)-supermagic labeing.

Proof. \(n \equiv 0 (\text{mod}\ 2)\)
The labeling \(h_0\) is defined as: \begin{align*} h_0(v_1)&=1,\\ h_0(v_2)&=\frac{n}{2}+2,\\ h_0(v_1v_2)&= 3n+3,\\ h_0(v_1x_i)&=3n+3-i.\\ \end{align*} \[ h_0(x_{i})= \begin{cases} \frac{n}{2}+2-i \ \ & \ \ \ \ \ \textrm{ if $i = 1,2,\dots, \frac{n}{2}$} \\ \frac{3n+6}{2}-i \ \ & \ \ \ \ \ \textrm{ if $i = \frac{n}{2}+1,\frac{n}{2}+2, \dots, n$} \\ \end{cases} \] \[ h_0(v_2x_{i})= \begin{cases} n+2(1+i) \ \ & \textrm{ if $i = 1,2,\dots, \frac{n}{2}$} \\ 2i+1 \ \ & \textrm{ if $i = \frac{n}{2}+1,\frac{n}{2}+2, \dots, n$} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_0\) and edges receive the labels \(\{n+3, n+4,\dots, 3n+3\}\). Therefore \(h_0\) is a super total labeling.
Using equation (1)

When \(n \equiv 1 \ \ (\text{mod}\ 2)\)
The labeling \(h_0\) is defined as: \begin{align*} h_0(v_i)&=i,\\ h_0(v_1v_2)&= n+3,\\ h_0(x_i)&=n+3-i.\\ \end{align*} For \(i \equiv 0 \) (mod \(2\)) \[ h_0(v_jx_{i})= \begin{cases} n+3 +\frac{i}{2} \ \ & \textrm{ if $j = 1$} \\ \frac{5n+7+i}{2} \ \ & \textrm{ if $j = 2$} \\ \end{cases} \] For \(i \equiv 1\) (mod \(2\)) \[ h_0(v_jx_{i})= \begin{cases} \frac{3(n+2)+i}{2} \ \ & \textrm{ if $j = 1$} \\ \frac{4n+7+i}{2} \ \ & \textrm{ if $j = 2$} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_0\) and edges receive the labels \(\{n+3, n+4,\dots, 3n+3\}\). Therefore \(h_0\) is a super total labeling.
Using equation (1) Equations (2, 3) shows \(wt_{h_0}(C_3^{(i)})\) is independent of \(i\). Hence the corona graph \(G\) admits a \(C_3\)-supermagic labeling. This completes the proof.

2.2. Super \((a, d)\)-\(C_3\)-antimagic labeling

Theorem 2.2. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a super \((a,1)\)-\(C_3\)-antimagic labeing.

Proof. The labeling \(h_1\) is defined as: \begin{align*} h_1(v_i)&=i,\\ h_1(v_1v_2)&= n+3,\\ h_1(v_2x_{i})&= 2n+3+i. \end{align*} \[ h_1(x_{i})= \begin{cases} \frac{i+1}{2}+2 \ \ & \textrm{ if $i \equiv 1$ (mod \ $2$)} \\ \lceil\frac{n}{2}\rceil+ 2 +\frac{i}{2} \ \ & \textrm{ if $i \equiv 0$ (mod $2$)} \\ \end{cases} \] \[ h_1(v_1x_{i})= \begin{cases} \frac{4n+7-i}{2} \ \ & \textrm{ if $i \equiv 1$ (mod \ $2$)} \\ \lceil\frac{n-1}{2}\rceil+ n+4-\frac{i}{2} \ \ & \textrm{ if $i \equiv 0$ (mod $2$)} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_1\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_1\) is a super total labeling.
Using equation (1)

Equation (4) shows \(wt_{h_0}(C_3^{(i)})\) constitute an arithmetic progression with \(a=5(n+3)+1\) and \(d=1\). Hence the corona graph \(G\) admits a super \((a,1)\)-\(C_3\)-antimagic labeling. This completes the proof.

Theorem 2.3. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a super \((a,d)\)-\(C_3\)-antimagic labeing for \(d=3,5\).

Proof. The labeling \(h_d\) is defined as: \begin{align*} h_d(v_i)&=i,\\ h_d(v_1v_2)&= n+3,\\ h_d(x_i)&= 2+i. \end{align*} \[ h_3(v_jx_{i})= \begin{cases} 2n+3+i \ \ & \textrm{ if $j=1$} \\ n+3+i \ \ & \textrm{ if $j=2$} \\ \end{cases} \] \[ h_5(v_jx_{i})= \begin{cases} n+2+2i \ \ & \textrm{ if $j=1$} \\ n+3+2i \ \ & \textrm{ if $j=2$} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_d\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_d\) is a super total labeling.
Using equation (1)

Equation (5) shows \(wt_{h_3}(C_3^{(i)})\) constitute an arithmetic progression with \(a=2(2n+7)+3\) and \(d=3\). Hence the corona graph \(G\) admits a super \((a,3)\)-\(C_3\)-antimagic labeling.
Now, for case \(d=5\), Using equation (1) Equation (6) shows \(wt_{h_3}(C_3^{(i)})\) constitute an arithmetic progression with \(a=3(n+6)\) and \(d=5\). Hence the corona graph \(G\) admits a super \((a,5)\)-\(C_3\)-antimagic labeling. This completes the proof.

Theorem 2.4. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a super \((a,d)\)-\(C_3\)-antimagic labeing for \(d=2,4\).

Proof. The labeling \(h_d\) is defined as: $$h_d(v_i)=i$$ \[ h_d(x_{i})= \begin{cases} n+3-i \ \ & \textrm{ if $d=2$} \\ 2+i \ \ & \textrm{ if $d=4$} \\ \end{cases} \] The edges are labeled as:
When \(n \equiv 0 \ \ \ (\text{mod} \;2)\)
$$h_d(v_1v_2)= 5\left(\frac{n}{2}\right)+3$$ \[ h_d(v_1x_{i})= \begin{cases} n + 2 + i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n}{2}+1$ }\\ \frac{n}{2}+1+2i \ \ & \textrm{ if $i = \frac{n}{2} +2, \frac{n}{2} +3,...,n$}\\ \end{cases} \] \[ h_d(v_2x_{i})= \begin{cases} \frac{3n}{2}+ 2(1 + i) \ \ & \textrm{ if $i = 1,2,\dots,\frac{n}{2}+1$ }\\ 2n+3+i \ \ & \textrm{ if $i = \frac{n}{2}+2, \frac{n}{2}+3,...,n$}\\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_d\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_d\) is a super total labeling.
Using equation (1)

Equation (7) shows \(wt_{h_2}(C_3^{(i)})\) constitute an arithmetic progression with \(a=6n+15\) and \(d=2\). Hence the corona graph \(G\) admits a super \((a,2)\)-\(C_3\)-antimagic labeling. Using equation (1) Equation (8) shows \(wt_{h_4}(C_3^{(i)})\) constitute an arithmetic progression with \(a=5n+16\) and \(d=4\). Hence the corona graph \(G\) admits a super \((a,4)\)-\(C_3\)-antimagic labeling.
When \(n \equiv 1 \ \ (\text{mod}\; 2)\)
$$h_d(v_1v_2)= 3n+3$$ \[ h_d(v_1x_{i})= \begin{cases} n + 2 + i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n+1}{2}$}\\ \frac{n+1}{2} + 1 + 2i \ \ & \textrm{ if $i = \frac{n+1}{2}+1, \frac{n+1}{2} +2,...,n$}\\ \end{cases} \] \[ h_d(v_2x_{i})= \begin{cases} \frac{n+1}{2}+n+1+2i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n+1}{2}$}\\ 2(n+1)+i \ \ & \textrm{ if $i = \frac{n+1}{2}+1, \frac{n+1}{2}+2,...,n$}\\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_d\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_d\) is a super total labeling.
Using equation (1) Equation (9) shows \(wt_{h_2}(C_3^{(i)})\) constitute an arithmetic progression with \(a=\frac{13n+29}{2}\) and \(d=2\). Hence the corona graph \(G\) admits a super \((a,2)\)-\(C_3\)-antimagic labeling.
Using equation (1) Equation (10) shows \(wt_{h_4}(C_3^{(i)})\) constitute an arithmetic progression with \(a=\frac{11n+31}{2}\) and \(d=4\). Hence the corona graph \(G\) admits a super \((a,4)\)-\(C_3\)-antimagic labeling. This completes the proof.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Integral inequalities and their extensions have received considerable attention in the theory of differential and difference equations. Recently, there has been much interest in the study of integral inequalities on time scales. Researchers have studied various aspects of inequalities on time scales [1, 2, 3]. New researches on dynamic inequalities using time scales was done by Agarwal in a monograph 4.

The fractional calculus is an extensions of derivatives and integrals to noninteger orders. This subject came to the attention of many researchers and fractional calculus on time scales has been studied [5]. In [6], some fractional integral inequalities have been studied in the real case. For more investigations, we refer [7, 8] to the readers.

In this paper, we obtain some generalizations and refinements for some existing inequalities on time scales. We consider the functional [9] \begin{equation*} \ T\left( f,g\right) :=\frac{1}{t-a}\int_{a}^{t}f\left( x\right) g\left( x\right) \Delta x-\frac{1}{t-a}\left( \int_{a}^{t}f\left( x\right) \Delta x\right) \left( \frac{1}{t-a}\int_{a}^{t}g\left( x\right) \Delta x\right) \end{equation*} where \(f\) and \(g\) are two integrable functions which are synchronous on \(\left[ a,t\right] _{\mathbb{T}}.\) (i.e. \(\left( f\left( x\right) -f\left( y\right) \right) \left( g\left( x\right) -g\left( y\right) \right) \geq 0~for~any~x,y\in \left[ a,t\right] _{\mathbb{T}}.)~\) The intervals with \(\mathbb{T}\) subscript are used to denote the intersection of the usual interval with \(\mathbb{T}\); i.e., \(\left[ a,t\right]_{\mathbb{T}}:=\left[ a,t\right] \cap \) \(\mathbb{T}\) for convenience.

The main purpose of this paper is to establish some new fractional inequalities for synchronous functions using delta Reimann-Liouville fractional integrals on time scales. Our results unify and extend some continuous inequalities and their corresponding discrete analogues. In the following, we present some basic concepts about time scale calculus and refer the reader to resource [7] for more detailed information on subject.

2. Preliminaries

A time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of the real numbers \(\mathbb{R}\). For \(t\in \mathbb{T}\) we define the forward jump operator \(\sigma :\mathbb{T\rightarrow T}\) by \begin{equation*} \sigma \left( t\right) :=\inf \left\{ s\in \mathbb{T}\text{:~}s>t\right\} \end{equation*} while the backward jump operator \(\rho:\mathbb{T\rightarrow T}\) is defined by \begin{equation*} \rho \left( t\right) :=\sup \left\{ s\in \mathbb{T}\text{:~}s< t\right\} . \end{equation*} If \(\sigma \left( t\right) >t\), we say that \(t\) is \textit{right-scattered}, while if \(\rho \left( t\right) < t~\) we say that \(t\) is left-scattered. Also, if \(\sigma \left( t\right) =t\), then \(t\) is called right-dense, and if \(\rho \left( t\right) =t,\) then \(t\) is called left-dense. The graininess function \(\mu :\mathbb{T\rightarrow }\left[0,\infty \right)\) is defined by \begin{equation*} \mu \left( t\right) :=\sigma \left( t\right)-t. \end{equation*} We introduce the set \(\mathbb{T}^{\kappa}\) which is derived from the time scale \(\mathbb{T}\) as follows. If \(\mathbb{T}\) has left-scattered maximum \(m, \) then \(\mathbb{T}^{\kappa}=\mathbb{T-}\left\{ m\right\} ,\) otherwise \(\mathbb{T}^{\kappa }=\mathbb{T}\).

Definition 2.1. Let \(f\) and \(g\) be two integrable functions defined on \(\left[ a,t\right] _{\mathbb{T}}\). If for any \( x,y\in \left[ a,t\right] _{\mathbb{T}} \) \begin{equation*} \ \left[ f\left( x\right) -f\left( y\right) \right] \left[ g\left( x\right) -g\left( y\right) \right] \geq 0, \end{equation*} then \(f\) and \(g\) are called synchronous functions on \(\left[ a,t\right] _{\mathbb{T}}.\)

Definition 2.2. The rd continuous functions \(h_{\alpha}:\mathbb{T\times T\rightarrow R}\) is called as generalized polynomials on time scales such that, for all \(s,t\in \mathbb{T}\) and \(\alpha \geq 0\); \begin{eqnarray*} h_{0}\left( t,s\right) &=&1 \\ h_{\alpha +1}\left( t,s\right) &=&\int_{s}^{t}h_{\alpha }\left( \tau ,s\right) \Delta \tau \end{eqnarray*}

Definition 2.3. The function \(f:\mathbb{T}\rightarrow \mathbb{R}\) is called rd-continuous provided it is continuous at right-dense points in \(\mathbb{T}\) and its left-sided limits exist (finite) at left-dense points in \(\mathbb{T}\).

Definition 2.4. The Delta-Riemann-Liouville fractional integral operator of order \(\alpha \geq 1\) on time scales, for a function \(f\in C_{rd}\) is defined as \begin{eqnarray*} D_{a}^{\alpha }f\left( t\right) &=&\int_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \tau \right) \Delta \tau , \\ D_{a}^{0}f &=&f. \end{eqnarray*}

3. Main Results

In this section we present our main results.

Theorem 3.1. Let \(f\) and \(g\) be two synchronous functions on \( \left[ 0,\infty \right) _{\mathbb{T}}\). Then for all \(t>a\), \(\alpha \geq 1\), \( a\geq 0\) we have

Proof. Since \(f\)and \(g\) are synchronous functions on \(\left[ 0.\infty \right) _{\mathbb{T}}\), then for all \(\tau ,\phi \geq 0\),

and Therefore, For \(\tau \in \left( a,t\right)\) , multiplying both sides of (4) by \(h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right),\) we have Integrating (5) over \(\left( a,t\right)\), we get \begin{eqnarray*} &&\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \tau \right) g\left( \tau \right) \Delta \tau +\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \phi \right) g\left( \phi \right) \Delta \tau \\ &&\geq \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \tau \right) g\left( \phi \right) \Delta \tau +\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \phi \right) g\left( \tau \right) \Delta \tau . \end{eqnarray*} Since \(f\left( \phi \right) ,g\left( \phi \right)\) and \(f\left( \phi \right) g\left( \phi \right)\) are independent from \(\tau\), we take them out of the integral and we obtain \begin{eqnarray*} &&\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \tau \right) g\left( \tau \right) \Delta \tau +f\left( \phi \right) g\left( \phi \right) \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) \Delta \tau \\ &&\geq g\left( \phi \right) \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \tau \right) \Delta \tau +f\left( \phi \right) \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) g\left( \tau \right) \Delta \tau. \end{eqnarray*} Using Definition 2.2 and Definition 2.4, we find For \(\phi \in \left( a,t\right),\) multiplying both sides of (6) by \( h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right)\) , we obtain Integrating (7) over \(\left( a,t\right)\), we get \begin{eqnarray*} &&\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) D_{a}^{\alpha }\left( fg\right) \left( t\right) \Delta \phi +\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) f\left( \phi \right) g\left( \phi \right) \left( h_{\alpha }\left( t,a\right) \right) \Delta \phi \\ &&\geq \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) g\left( \phi \right) D_{a}^{\alpha }f\left( t\right) \Delta \phi +\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) f\left( \phi \right) D_{a}^{\alpha }g\left( t\right) \Delta \phi . \end{eqnarray*} Since \(D_{a}^{\alpha }\left( fg\right) \left( t\right)\), \(D_{a}^{\alpha }f\left( t\right) ,D_{a}^{\alpha }g\left( t\right)\) and \(h_{\alpha }\left( t,a\right)\) are independent from \(\phi\), we obtain \begin{eqnarray*} &&D_{a}^{\alpha }\left( fg\right) \left( t\right) \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) \Delta \phi +\left( h_{\alpha }\left( t,a\right) \right) \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) f\left( \phi \right) g\left( \phi \right) \Delta \phi \\ &&\geq D_{a}^{\alpha }f\left( t\right) \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) g\left( \phi \right) \Delta \phi +D_{a}^{\alpha }g\left( t\right) \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \phi \right) \right) f\left( \phi \right) \Delta \phi . \end{eqnarray*} Therefore from Definition 2.2 and Definition 2.4, we can write \begin{eqnarray*} &&D_{a}^{\alpha }\left( fg\right) \left( t\right) \left( h_{\alpha }\left( t,a\right) \right) +\left( h_{\alpha }\left( t,a\right) \right) D_{a}^{\alpha }\left( fg\right) \left( t\right) \\ &&\geq D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) +D_{a}^{\alpha }g\left( t\right) D_{a}^{\alpha }f\left( t\right) . \end{eqnarray*} So we have \begin{equation*} 2D_{a}^{\alpha }\left( fg\right) \left( t\right) \left( h_{\alpha }\left( t,a\right) \right) \geq 2D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) , \end{equation*} \begin{equation*} D_{a}^{\alpha }\left( fg\right) \left( t\right) \left( h_{\alpha }\left( t,a\right) \right) \geq D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) , \end{equation*} and finally we get \begin{equation*} D_{a}^{\alpha }\left( fg\right) \left( t\right) \geq \frac{1}{\left( h_{\alpha }\left( t,a\right) \right) }D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) . \end{equation*}

Theorem 3.2. Let \(f\) and \(g\) be two synchronous functions on \(\left[ 0,\infty \right) _{\mathbb{T}}\). Then for all \(t>a\), \(\alpha ,\beta \geq 1\), \(a\geq 0\) we have: \begin{equation*} h_{\alpha }\left( t,a\right) D_{a}^{\beta }\left( fg\right) \left( t\right) +h_{\beta }\left( t,a\right) D_{a}^{\alpha }\left( fg\right) \left( t\right) \geq D_{a}^{\alpha }f\left( t\right) D_{a}^{\beta }g\left( t\right) +D_{a}^{\alpha }g\left( t\right) D_{a}^{\beta }f\left( t\right) . \end{equation*}

Proof. Since \(f\) and \(g\) are synchronous functions on \(\left[0.\infty \right) _{\mathbb{T}}\), then for all \(\tau ,\phi \geq 0\), we have \begin{equation*} \left( f\left( \tau \right) -f\left( \phi \right) \right) \left( g\left( \tau \right) -g\left( \phi \right) \right) \geq 0. \end{equation*} Therefore,

For \(\tau \in \left( a,t\right),\) multiplying both sides of (8) by \(h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right)\), we obtain Integrating (9) over \(\left( a,t\right)\), we get \begin{eqnarray*} &&\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \tau \right) g\left( \tau \right) \Delta \tau +\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \phi \right) g\left( \phi \right) \Delta \tau \\ &&\geq \int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \tau \right) g\left( \phi \right) \Delta \tau +\int\limits_{a}^{t}h_{\alpha -1}\left( t,\sigma \left( \tau \right) \right) f\left( \phi \right) g\left( \tau \right) \Delta \tau . \end{eqnarray*} Consequently For \(\phi \in \left( a,t\right),\) multiplying both sides of (10) by \(h_{\beta -1}\left( t,\sigma \left( \phi \right) \right)\), we have Integrating (11) over \(\left( a,t\right)\), we get \begin{eqnarray*} &&\int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) D_{a}^{\alpha }\left( fg\right) \left( t\right) \Delta \phi +\int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) \left( fg\right) \left( \phi \right) \left( h_{\alpha }\left( t,a\right) \right) \Delta \phi \\ &&\geq \int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) g\left( \phi \right) D_{a}^{\alpha }f\left( t\right) \Delta \phi +\int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) f\left( \phi \right) D_{a}^{\alpha }g\left( t\right) \Delta \phi. \end{eqnarray*} then \begin{eqnarray*} &&D_{a}^{\alpha }\left( fg\right) \left( t\right) \int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) \Delta \phi +\left( h_{\alpha }\left( t,a\right) \right) \int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) f\left( \phi \right) g\left( \phi \right) \Delta \phi \\ &&\geq D_{a}^{\alpha }f\left( t\right) \int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) g\left( \phi \right) \Delta \phi +D_{a}^{\alpha }g\left( t\right) \int\limits_{a}^{t}h_{_{\beta -1}}\left( t,\sigma \left( \phi \right) \right) f\left( \phi \right) \Delta \phi . \end{eqnarray*} Therefore, we write \begin{equation*} D_{a}^{\alpha }\left( fg\right) \left( t\right) \left( h_{_{\beta }}\left( t,a\right) \right) +\left( h_{\alpha }\left( t,a\right) \right) D_{a}^{\beta }\left( fg\right) \left( t\right) \geq D_{a}^{\alpha }f\left( t\right) D_{a}^{\beta }g\left( t\right) +D_{a}^{\alpha }g\left( t\right) D_{a}^{\beta }f\left( t\right) . \end{equation*}

Theorem 3.3. Let \(\left( f_{i}\right) _{=1,\ldots ,n}\) be \(n\) positive increasing functions on \(\left[ 0,\infty \right)_{\mathbb{T}}\). Then for all \(t>a\), \(\alpha \geq 1\), \(a\geq 0\) we have \begin{equation*} D_{a}^{\alpha }\left( \prod_{i=1}^{n}f_{i}\right) \left( t\right) \geq \left( h_{\alpha }\left( t,a\right) \right) ^{1-n}\prod_{i=1}^{n}D_{a}^{\alpha }f_{i}\left( t\right) . \end{equation*}

Proof. We use induction method to prove our result. Clearly, for \(n=1\), all \(t>a\), \(\alpha \geq 1,\) \(a\geq 0\) we have \begin{equation*} D_{a}^{\alpha }\left( f_{1}\right) \left( t\right) \geq D_{a}^{\alpha }f_{1}\left( t\right). \end{equation*} For \(n=2\), applying Theorem 3.1, for all \(t>a\), \(\alpha \geq 1\), \(a\geq 0\) we obtain \begin{equation*} D_{a}^{\alpha }\left( \prod_{i=1}^{2}f_{i}\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\prod_{i=1}^{2}D_{a}^{% \alpha }f_{i}\left( t\right) \end{equation*} then, \begin{equation*} D_{a}^{\alpha }\left( f_{1}f_{2}\right) \left( t\right) \geq \frac{1}{% h_{\alpha }\left( t,a\right) }D_{a}^{\alpha }f_{1}\left( t\right) D_{a}^{\alpha }f_{2}\left( t\right). \end{equation*} Now for \(n-1\), we assume that (induction hypothesis) the following inequality \begin{equation} D_{a}^{\alpha }\left( \prod_{i=1}^{n-1}f_{i}\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{2-n}\prod_{i=1}^{n-1}D_{a}^{% \alpha }f_{i}\left( t\right) \label{13} \end{equation} holds. We have to prove that the inequality \begin{equation*} D_{a}^{\alpha }\left( \prod_{i=1}^{n}f_{i}\left( t\right) \right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{1-n}\prod_{i=1}^{n}D_{a}^{% \alpha }f_{i}\left( t\right) \end{equation*} holds for \(n\). Since \(\left( f_{i}\right) _{i=1,2,...,n}\) are positive increasing functions, then \(\left( \prod_{i=1}^{n-1}f_{i}\right) \left( t\right)\) is an increasing function. Hence we can apply Theorem 3.1 to the functions \(\prod_{i=1}^{n-1}f_{i}=g\), \(f_{n} =f\). We obtain \begin{equation*} \prod_{i=1}^{n}f_{i}=\prod_{i=1}^{n-1}f_{i}f_{n}=fg \end{equation*} \begin{equation*} D_{a}^{\alpha }\left( fg\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }\left( f\right) \left( t\right) D_{a}^{\alpha }\left( g\right) \left( t\right) \end{equation*} \begin{equation*} D_{a}^{\alpha }\left( \prod_{i=1}^{n}f_{i}\right) \left( t\right) =D_{a}^{\alpha }\left( \prod_{i=1}^{n-1}f_{i}f_{n}\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }\left( \prod_{i=1}^{n-1}f_{i}\right) \left( t\right) D_{a}^{\alpha }\left( f_{n}\right) \left( t\right) . \end{equation*} Multiplying both sides of (12) by \(\left[ h_{\alpha }\left(t,a\right) \right] ^{-1}D_{a}^{\alpha }\left( f_{n}\right) \left( t\right) \) , we obtain \begin{eqnarray*} &&\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }\left( f_{n}\right) \left( t\right) D_{a}^{\alpha }\left( \prod_{i=1}^{n-1}f_{i}\right) \left( t\right) \\&&\geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\left[ h_{\alpha }\left( t,a\right) \right] ^{2-n}\prod_{i=1}^{n-1}D_{a}^{\alpha }f_{i}\left( t\right) D_{a}^{\alpha }f_{n}\left( t\right) \end{eqnarray*} and% \begin{equation*} \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }\left( f_{n}\right) \left( t\right) D_{a}^{\alpha }\left( \prod_{i=1}^{n-1}f_{i}\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{1-n}\prod_{i=1}^{n}D_{a}^{\alpha }f_{i}\left( t\right). \end{equation*} Therefore we obtain \begin{equation*} D_{a}^{\alpha }\left( \prod_{i=1}^{n}f_{i}\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{1-n}\prod_{i=1}^{n}D_{a}^{% \alpha }f_{i}\left( t\right). \end{equation*}

Theorem 3.4. Let \(f\) and \(g\) be two functions defined on \(\left[ 0,\infty \right) _{\mathbb{T}}\) , such that \(f\) is increasing and \(g\) is differentiable and there exist a real number \(m:=\inf_{t\geq 0}g^{'}\left( t\right)\). Then for all \(t>a\), \(\alpha \geq 1,\) \(a\geq 0\) we have \begin{eqnarray*} D_{a}^{\alpha }\left( fg\right) \left( t\right)& \geq& \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -m\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( t\right) \\&&+mD_{a}^{\alpha }\left( tf\left( t\right) \right). \end{eqnarray*}

Proof. We consider the function \(k\left( t\right) :=g\left( t\right) -mt\). It is evident that \(k\) is differentiable and it is increasing on \(\left[ 0,\infty \right)_{\mathbb{T}}\). Then applying the Theorem 3.1 we have \begin{equation*} D_{a}^{\alpha }\left( kf\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }k\left( t\right), \end{equation*} then \begin{equation*} D_{a}^{\alpha }\left( \left( g-mt\right) f\left( t\right) \right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( g-mt\right) \left( t\right). \end{equation*} So we get \begin{gather*} D_{a}^{\alpha }\left( fg\right) \left( t\right) -mD_{a}^{\alpha }\left( tf\left( t\right) \right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) \left( D_{a}^{\alpha }g\left( t\right) -mD_{a}^{\alpha }\left( t\right) \right) \\ =\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -m\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( t\right), \end{gather*} and we obtain our desired inequality.

Corollary 3.5. Let \(f\) and \(g\) be two functions defined on \(\left[ 0,\infty \right) _{\mathbb{T}}\).
(1). Assume that \(f\) is decreasing, \(g\) is differentiable and there exist a real number \(M:=\sup_{t\geq 0}g^{'}\left( t\right)\). Then for all \(t>a\), \(\alpha \geq 1\), \(a\geq 0\), we have \begin{eqnarray*} D_{a}^{\alpha }\left( fg\right) \left( t\right) &\geq& \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -M\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( t\right)\\&& +MD_{a}^{\alpha }\left( tf\left( t\right) \right). \end{eqnarray*} (2). Assume that \(f\) and \(g\) are differentiable and there exist \(m_{1}:=\inf_{t\geq 0}f^{'}\left( t\right)\), \(m_{2}:=\inf_{t\geq 0}g^{'}\left( t\right)\). Then for all \(t>a\), \(\alpha \geq 1, a\geq 0\), we have \begin{eqnarray*} &&D_{a}^{\alpha }\left( fg\right) \left( t\right) -m_{2}D_{a}^{\alpha }\left( tf\left( t\right) \right) -m_{1}D_{a}^{\alpha }\left( tg\left( t\right) \right) +m_{1}m_{2}D_{a}^{\alpha }t^{2} \\ &&\geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\big[ D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -m_{2}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( t\right) -m_{1}D_{a}^{\alpha }\left( t\right) D_{a}^{\alpha }g\left( t\right) \\&&+m_{1}m_{2}\left( D_{a}^{\alpha }\left( t\right) \right) ^{2}\big]. \end{eqnarray*} (3). Assume that \(f\) and \(g\) are differentiable and there exist \(M_{1}:=\sup_{t\geq 0}f^{'}\left( t\right)\), \(M_{2}:=\sup_{t\geq 0}g^{'}\left( t\right)\). Then for all \(t>a\), \(\alpha \geq 1\), \( a\geq 0\), we have \begin{eqnarray*} &&D_{a}^{\alpha }\left( fg\right) \left( t\right) -M_{2}D_{a}^{\alpha }\left( tf\left( t\right) \right) -M_{1}D_{a}^{\alpha }\left( tg\left( t\right) \right) +M_{1}M_{2}D_{a}^{\alpha }t^{2} \\ &&\geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\big[ D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -M_{2}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( t\right) -M_{1}D_{a}^{\alpha }\left( t\right) D_{a}^{\alpha }g\left( t\right) \\&&+M_{1}M_{2}\left( D_{a}^{\alpha }\left( t\right) \right) ^{2}\big]. \end{eqnarray*}

Proof. (1). Let \(G\left( t\right) :=g\left( t\right) -Mt\) be a function such that it is decreasing and differentiable on \(\left[ 0,\infty \right) _{\mathbb{T}}\). Then for all \(t>a\), \(\alpha \geq 1\), \(a\geq 0,\) we have \begin{equation*} D_{a}^{\alpha }\left( Gf\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }G\left( t\right) D_{a}^{\alpha }f\left( t\right) \end{equation*} by Theorem 3.1. Hence, we get \begin{equation*} D_{a}^{\alpha }\left( \left( g-Mt\right) f\left( t\right) \right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( g-Mt\right) \left( t\right). \end{equation*} Then we have \begin{gather*} D_{a}^{\alpha }\left( fg\right) \left( t\right) -MD_{a}^{\alpha }\left( tf\left( t\right) \right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) \left( D_{a}^{\alpha }g\left( t\right) -MD_{a}^{\alpha }\left( t\right) \right) \\ =\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -M\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( t\right), \end{gather*} and so \begin{eqnarray*} D_{a}^{\alpha }\left( fg\right) \left( t\right)& \geq &\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -M\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }\left( t\right)\\&& +MD_{a}^{\alpha }\left( tf\left( t\right) \right). \end{eqnarray*} (2). Let \(F\left( t\right) :=f\left( t\right) -m_{1}t\), \(G\left( t\right) :=g\left( t\right) -m_{2}t\) be two functions such that they are increasing and differentiable on \(\left[ 0,\infty \right) _{\mathbb{T}}\). Then for all \(t>a\), \(\alpha \geq 1\), \(a\geq 0\) and by Theorem 3.1 we have \begin{equation*} D_{a}^{\alpha }\left( FG\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }F\left( t\right) D_{a}^{\alpha }G\left( t\right), \end{equation*} and \begin{eqnarray*} &&D_{a}^{\alpha }\left( \left( f\left( t\right) -m_{1}t\right) \left( g\left( t\right) -m_{2}t\right) \right) \left( t\right)\\&& \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }\left( f\left( t\right) -m_{1}t\right) \left( t\right) D_{a}^{\alpha }\left( g\left( t\right) -m_{2}t\right) \left( t\right). \end{eqnarray*} Then we get \begin{eqnarray*} &&D_{a}^{\alpha }\left( \left( fg\right) \left( t\right) -m_{2}tf\left( t\right) -m_{1}tg\left( t\right) +m_{1}m_{2}t^{2}\right) \\&& \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\left[ D_{a}^{\alpha }f\left( t\right) -m_{1}D_{a}^{\alpha }t\right] \left[ D_{a}^{\alpha }g\left( t\right) -m_{2}D_{a}^{\alpha }t\right] \\&& =\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\big[ D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -m_{2}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }t-m_{1}D_{a}^{\alpha }g\left( t\right) D_{a}^{\alpha }t\\&&+m_{1}m_{2}\left( D_{a}^{\alpha }t\right) ^{2}\big]. \end{eqnarray*} Therefore we obtain \begin{eqnarray*} &&D_{a}^{\alpha }\left( \left( fg\right) \left( t\right) -m_{2}tf\left( t\right) -m_{1}tg\left( t\right) +m_{1}m_{2}t^{2}\right) \\ &&\geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\big[ D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -m_{2}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }t-m_{1}D_{a}^{\alpha }g\left( t\right) D_{a}^{\alpha }t\\&&+m_{1}m_{2}\left( D_{a}^{\alpha }t\right) ^{2}\big]. \end{eqnarray*} (3). Let \(F\left( t\right) :=f\left( t\right) -M_{1}t\), \( G\left( t\right) :=g\left( t\right) -M_{2}t\) be two functions such that they are decreasing and differentiable on \(\left[ 0,\infty \right) _{\mathbb{T}}\). Then for all \(t>a\), \(\alpha \geq 1\), \(a\geq 0\) and by Theorem 3.1, we have \begin{eqnarray*} D_{a}^{\alpha }\left( FG\right) \left( t\right) \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }F\left( t\right), D_{a}^{\alpha }G\left( t\right)\end{eqnarray*} so \begin{eqnarray*} && D_{a}^{\alpha }\left( \left( f\left( t\right) -M_{1}t\right) \left( g\left( t\right) -M_{2}t\right) \right) \left( t\right)\\&& \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}D_{a}^{\alpha }\left( f\left( t\right) -M_{1}t\right) \left( t\right) D_{a}^{\alpha }\left( g\left( t\right) -M_{2}t\right) \left( t\right). \end{eqnarray*} Hence we obtain \begin{eqnarray*}&& D_{a}^{\alpha }\left( \left( fg\right) \left( t\right) -M_{2}tf\left( t\right) -M_{1}tg\left( t\right) +M_{1}M_{2}t^{2}\right) \\&& \geq \left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\left[ D_{a}^{\alpha }f\left( t\right) -M_{1}D_{a}^{\alpha }t\right] \left[ D_{a}^{\alpha }g\left( t\right) -M_{2}D_{a}^{\alpha }t\right] \\&& =\left[ h_{\alpha }\left( t,a\right) \right] ^{-1}\big[ D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }g\left( t\right) -M_{2}D_{a}^{\alpha }f\left( t\right) D_{a}^{\alpha }t-M_{1}D_{a}^{\alpha }g\left( t\right) D_{a}^{\alpha }t\\&&+M_{1}M_{2}\left( D_{a}^{\alpha }t\right) ^{2}\big]. \end{eqnarray*}

4. Conclusion

In this paper, we have studied some fractional integral inequalities and have extended results for time scale calculus. The theorems in this work improves previously results and this presents a new approach to use new definitions of fractional integrals for integral inequalities on time scales

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Definition 1.1. A group \(G\) is said to be with finite conjugacy classes (or shortly \(FC\)-group) if and only if every element of \(G\) has a finite conjugacy class in \(G\).

It is known that \(FIZ\subseteq FA\subseteq FC\), where \(FIZ\) denotes the class of center-by-finite groups, and that for finitely generated equalities \(FIZ=FA=FC\) hold. These results and other have been studied and developed by Baer, Neumann, Erdos and Tomkinson and others in [1, 2, 3, 4, 5]. \(FC\)-groups have many similar properties with abelian groups and finite groups. It is known that the class of \(FC\)-groups is closed under taking subgroup, homomorphic images, quotient and forming restricted direct products, but it is not closed under taking finite extension. We prove in Theorem 2.2, that in the class of finitely generated \(\tau N\)-group (respectively \(FN\)-group) a \((FC)\tau \)-group (respectively \((FC)F\)-group) is \(\tau A\)-group (respectively is \(FA\)-group).

Definition 1.2. Let \(\chi \) is a given property of groups. A group \(G\) it is said to be in the class \((\chi ,\) \(\infty )\) (respectively \((\chi ,\infty)^{\ast }\)) if and only if every infinite subset \(X\) of \(G\) contains two distinct elements \(x\), \(y\) such that the subgroup \(< x\), \(y >\) (respectively \(< x\), \(x^{y}>\)) is a \(\chi\)-group. Note that if \(\chi\) is a subgroup closed class, then \(\chi \subset (\chi , \infty )\ \subset (\chi , \infty)^{\ast }\).

On the one hand, several authors have studied the class of \((\chi , \infty)\)-groups, where \(\chi \) is a given property of groups, with some conditions on these groups. The question that interests mathematicians is the following: If \(G\) is a group in the class \((\chi ,\infty )\) where \(\chi\) is a given property, then does \(G\) have a property in relation to the property \(\chi\). For example \(G\) has the property \(\chi \gamma\) or \(\gamma \chi\), etc. where \(\gamma\) is another group property, or in particular \(G\) is in the same class \(\chi\). For example, in 1976, Neumann in [6], has shown that a group is in the class \((A,\infty)\) if and only if it is \(FIZ\)-group. In 1981, Lennox and Wiegold in [7] proved that a finitely generated solvable group is in the class \((N,\infty )\) (respectively \((P,\infty )\), \((C_{o},\infty )\)) if and only if it is \(FN\), (respectively \(P\), \(C_{o}\)), where \(P\) and \(C_{o}\) respectively polycyclic and coherent groups.

In 2000, 2002 and 2005, Abdollahi and Trabelsi, proved in [8, 9, 10] that a finitely generated solvable group is in the class \((FN_{k}, \infty )\) (respectively \((FN, \infty)\), \((NF, \infty )\), \((\tau N, \infty )\)) if and only if it is \(FN_{k}^{(2)}\), (respectively \(FN,NF,\tau N\)). Other results of this type have been obtained, for example in [8, 9, 10, 11, 12, 13, 14, 15, 16]. In this note, we prove that a finitely generated \(\tau N\)-group \(G\) which is in the class \(((\tau N_{k})\tau ,\infty)\) is in the class \(\tau N_{k}^{(2)}\) and deduce that a finitely generated \(FN\)-group (respectively \(NF\)-group) \(G\) in the class of \(((FN_{k})F,\infty )\)-groups, is in the class of \(FN_{k}^{(2)}\)-groups (respectively in the class of \(N_{k}^{(2)}F\)-groups) and particularly a group \(G\) is in the class \(((FC)F,\infty )\) if and only if, it is \(FA\)-group.

On the other hand, in 2005, Trabelsi in [10](respectively in 2007, Rouabehi and Trabelsi in [17]) proved that a finitely generated soluble group in the class \((CN, \infty )^{\ast }\) where \(C\) is the class of cernikov group (respectively in the class in the class \((\tau N,\infty )^{\ast }\)) is \(FN\)-group (respectively \(\tau N\)-group) and in 2007 too, Guerbi and Rouabhi in [14] proved that a finitely generated Hyper(abelian-by-finite) group in the class \((\Omega ,\infty )^{\ast }\) where \(\Omega\) is the class of finite depth group, is \(FN\)-group. In this paper, we prove that a finitely generated \(\tau N\)-group in the class \(((FN_{k})\tau ,\infty )^{\ast }\) is in the class \(\tau N_{c}\) for certain integer \(c\) and deduce that a finitely generated \(FN\)-group (respectively \(NF\)-group) \(G\) in the class \(((FN_{k})F,\infty )^{\ast }\)) is in the class \(FN_{c}\) (respectively \(N_{c}F\)). In particular, if \(G\) is a finitely generated \(FN\)-group in the class \(((FC)F, \infty )^{\ast }\) (respectively \(((FN_{2})F, \infty )^{\ast }\)) then \(G\) is in the class of \(FN_{2}\)-groups (respectively in the class of \(FN_{3}^{(2)}\)-groups).

2. Main Results

2.1. Torsion and finite extension of property \(FC\)

It is known that the property \(FC\) is not closed under the formation of extension. The following example shows that even, a finite extension (respectively torsion extension) of an \(FC\)-group is not always an \(FC\)-group(respectively \(\tau A\)-group).

Note that if the center of an infinite finitely generated group is trivial or finite then this group is not \(FC\).

Example 2.1. Let \(G=D_{\infty }= < a,b/\) \( a^2 = 1 \) and \( aba =b^{-1} > \) the infinite dihedral group, which is a finitely generated soluble group, generated by the involutions \(a\), \(b\). We have \(K=C_{\infty }=< b >\) which is a infinite cyclic group isomorphic to \(\mathbb{Z}\) therefore it is a \(FC\)-group and the quotient group \(G/K\) is isomorphic to \(C_{2}=< a > \) which is finite of order \(2\), thus \(G\) is a finite extension of a \(FC\)-group, but as the center of the infinite dihedral group is trivial then it is not a \(FC\)-group.

This example shows also that \(D_{\infty }\) is a torsion extension of a \(FC\)-group but it is not a \(\tau A\)-group, so we consider the class of finitely generated \(\tau N\)-groups (respectively \(FN\)-group) and we prove that, in this class, a \((FC)\tau \)-group (respectively \((FC)F\)-group) is a \(\tau A\)-group (respectively \(FC\)-group).

Theorem 2.2. Let \(G\) a finitely generated torsion-by-nilpotent group. If \(G\) is \(FC\)-by-torsion group then \(G\) is \(\tau A\)-group.

Lemma 2.3. If \(G\) is a nilpotent group of nipotency class \(d\) and \(g\) an element of \(G.\) The subgroup \(< G^{\prime },g>\) generated by the derived group \(G^{\prime }\) and \(G\) is a nilpotent group of class \(\leq d\).

Lemma 2.4. If \(G\) is nilpotent and torsion-free group, \(m\), \(n\) two non-zero integers and \(x\),\(y\) \(\in G\), then,

1. If \(x^{n}=y^{n}\) then \(x=y\).
2. If \([x^{m}y^{n}]=1\) in \(G\), then \([x\ y]=1\) in \(G\).
3. If \([x^{m}yx^{n}y]=1\) in \(G\), then \([x\ y]=1\) in \(G\).

Proof. (1) We proceed by induction on the nipotency class \(d\) of the group \(G.\) If \(d=1\) so \(G\) is abelian: \(x^{n}= y^{n}\Longleftrightarrow x^{n}y^{-n}=1\Longleftrightarrow (xy^{-1})^{n}=1\) and as \(G\) is torsion-free then \(xy^{-1}=1\Longleftrightarrow x=y\). We suppose now that \(G\) is torsion-free nilpotent and non abelian of nilpotecy class \(d.\) We consider the subgroup \(H=< G^{\prime } x>\) generated by the derived group \(G^{\prime }\) and the element \(x\), by the Lemma 2.3 above the nilpotency class of \(H\) is less than \(d.\) Then by the inductive hypothesis the Lemma is verified for \(H.\) we have \(x\in H\) and \(x^{y}=y^{-1}xy=x[x y]\in H\) and \(x^{n}=y^{n}.\) So as \((y^{-1}xy)^{n}=y^{-1}x^{n}y=y^{-1}y^{n}y=y^{n}=x^{n}.\) The (1) in lemma applied to \(H\) give us that \(y^{-1}xy=x\) which means that \(x\) and \(y\) commute. So we have in \(G : x^{n}=y^{n}\Longleftrightarrow x^{n}y^{-n}=1\Longleftrightarrow (xy^{-1})^{n}=1\Longleftrightarrow x=y\).
(2) We have: \(x^{m}y^{n}=y^{n}x^{m}\Longleftrightarrow y^{-n}x^{m}y^{n}=x^{m}\Longleftrightarrow(y^{-n}xy^{n})^{m}=x^{m}\Longleftrightarrow y^{-n}xy^{n}=x\) (according to 1)\( \Longleftrightarrow xy^{n}=y^{n}x\Longleftrightarrow xy^{n}x^{-1}=y^{n}\Longleftrightarrow (xyx^{-1})^{n}=y^{n}\) also by (1) we obtain \(xyx^{-1}=y\) and so \(xy=yx.\)
(3) \([x^{m}y x^{n}y]=1\Longleftrightarrow x^{m}y x^{n}y=x^{n}y x^{m}y\Longleftrightarrow x^{m-n}y x^{n}=y x^{m}\Longleftrightarrow x^{m-n}y =y x^{m-n}\Longleftrightarrow \lbrack x^{m-n} y]=1\) by the (2) we obtain \(xy=yx\).

Proof of Theorem 2.2

Proof. Since \(G\) is finitely generated torsion-by-nilpotent group, there exists a normal and torsion subgroup \(F\) of \(G\) such that the quotient group \(G/F\) is nilpotent group. As the property \(FC\)-by-torsion is closed under quotient, it is enough to show that \(G/F\) is a \(FC\)-group. For this it is sufficient to show that every \(FC\)-by-torsion group \(G\) in the class of finitely generated nilpotent groups in a \(\tau A\)-group. Assume that \(G\) is \((FC)\)-by-torsion, so there exists a normal \(FC\)-subgroup \(N\) such that the quotient \(G/N\) is torsion. Since \(G\) is finitely generated and nilpotent, it checks the maximal condition on subgroups. So \(N\) is finitely generated \(FC\)-subgroup. According to ([1], Theorem 6.2 ) \(N\) is center-by-finite which means that \(Z(N)\) is of finite index in \(N\). Or \(G/N\) is torsion group. It follows that the quotient \(G/Z(N)\) is torsion group. So for all \(x\) and \(y\) in \(G\), there exist non-zero integers \(m\) and \(n\) such that \(x^{m}\) and \(y^{n}\) belong to \(Z(N)\), it follows that \([x^{m},y^{n}]=1\) in \(G\). Let \(T=Tor(G)\), the torsion subgroup of \(G\). Since \(G\) is nilpotent, the quotient group \(G/T\) is a finitely generated nilpotent torsion-free group, and therefore \([x^{m}T,y^{n}T]=T\) in \(G/T\). By the result (2) in Lemma 2.4 above, we deduce that \([xT,yT]=T\) in \(G/T\), which shows that the group \(G/T\) is an abelian group. More, since \(G\) is nilpotent and checks the maximal condition, by ([2], Theorem 5.1) the subgroup \(T\) as finitely generated nilpotent torsion group, is finite. Thus \(G\) is finite-by-Abelian so \(G\) is \(\tau A\). Since \(G/F\) is \(FC\)-by-torsion group in the class of finitely generated nilpotent-group, then \(G/F\) is \(\tau A\) and as \(F\) is torsion group it follows that \(G\) is \(\tau (\tau A)=\tau A\). This completes the proof.

Remark 2.1. The example below shows that Theorem 2.2. is falls when the condition "finitely generated" is omitted.

Example 2.5. Let \(A=F_{2}[X]\) algebra of polynomials on the field \(F_{2}\) and the isomorphism \(\varphi :A\times A\rightarrow A\times A\), \((P,Q)\mapsto (P+Q,Q)\). We put \(H=A\times A\) and \(K=< \varphi >\) such that \(\varphi^{2}=Id_{A\times A}\) the identity application on \(A\times A\). Since \(H\) is an abelian group, it is a \(FC\)-group. \(K\) is a finite group of order 2 and so it is \(FC\) too. We consider \(G=H\rtimes K\), the semi-direct product of \(H\) by \(K\). \(G\) is a non-finitely generated nilpotent group, which is a finite extension of the \(FC\)-group \(H\). But \(G\) is not a \(FC\)-group.

If we replace the property \(\tau\) by the property \(F\) we obtain a necessary and sufficient condition for the property \(FC\) to be closed under finite extension in the class of finitely generated \(FN\)-group.

Corollary 2.6. Let \(G\) a finitely generated finite-by-nilpotent group. \(G\) is \(FC\)-by-finite group if and only if \(G\) is \(FA\)-group.

Proof. It clear that if \(G\) is \(FA\)-group then \(G\) is \(FC\) and so \(FC\)-by-finite. If \(G\) is finitely generated finite-by-nilpotent group, as the same case in theorem above there exists a normal and finite subgroup \(F\) of \(G\) such that the quotient group \(G/F\) is nilpotent group. As the property \(FC\)-by-finite is closed under quotient, it is enough to show that \(G/F\) is a \(FC\)-group. For this it is sufficient to show that every \(FC\)-by-finite group \(G\) in the class of finitely generated nilpotent groups is a \(FA\)-group. Assume that \(G\) is \((FC)\)-by-finite, so there exists a normal \(FC\)-subgroup \(N\) such that the quotient \(G/N\) is finite. As the same way in the above theorem, we found that \(Z(N)\) is of finite index in \(N\) and the quotient \(G/Z(N)\) is finite group. So for all \(x\) and \(y\) in \(G\), there exist non-zero integers \(m \) and \(n\) such that \(x^{m}\) and \(y^{n}\) belong to \(Z(N)\), it follows that \([x^{m},y^{n}]=1\) in \(G\). If \(T=Tor(G)\), we have \([x^{m}T, y^{n}T]= T\) in \(G/T\). By the result (2) in Lemma 2.3, we deduce that \([xT, yT]=T\) in \(G/T\), which shows that the group \(G/T\) is an Abelian group.
Moreover, as in the above theorem we found that \(T\) is finite. Thus \(G\) is finite-by-Abelian. Since \(G/F\) is \(FC\)-by-finite group in the class of finitely generated nilpotent-group, it is \(FA\) and so \(G\) is \(F(FA)=FA\). This completes the proof.

\(\tau N_{k}\) and \(FN_{k}\)-groups and conditions on infinite subsets

Our first elementary propositions below follows from a results in [8, 12] and [10].

Proposition 2.7. If \(G\) is a finitely generated finite-by-soluble group in the class \((FN_{k}, \infty),\) then \(G\) is in the class of \(FN_{k}^{(2)}\)-groups.

Proof. Suppose that \(G\) is finite-by-soluble, there exists finite normal subgroup \(N\) such that \(G/N\) is soluble. As the class of \((FN_{k},\infty )\)-group, is closed under taking quotient, then the quotient group \(G/N\) is a finitely generated soluble group in the class of \((FN_{k},\infty )\)-group. By ([8] Corollary 1.8), \(G/N\) is in the class of \(FN_{k}^{(2)}\)-groups. Therefore \(G\) is finite-by-\(FN_{k}^{(2)}\)-group, and this gives that \(G\) is \(FN_{k}^{(2)}\)-group.

Proposition 2.8. If \(G\) is a finitely generated torsion-by-soluble group in the class \((\tau N_{k},\infty ),\) then \(G\) is in the class of \(\tau N_{k}^{(2)}\)-groups.

Proof. Suppose that \(G\) is finite-by-soluble, there exists a torsion and normal subgroup \(N\) such that \(G/N\) is soluble. As the class of \((\tau N_{k},\) \(\infty)\)-group, is closed under taking quotient, then the quotient group \(G/N\) is a finitely generated soluble group in the class of \((\tau N_{k},\infty )\subset (\tau N,\infty )\). By ([10], Theorem 1) \(G/N\) is in the class of \(\tau N\)-groups.
So \(G/N\) admits a torsion group \(\tau (G/N)=T/N\) such that \(T\) is torsion and the quotient \(G/T\) is torsion-free in the class \((\tau N_{k},\infty )\). So \(G/T\) is a finitely generated soluble group in the class \((N_{k}, \infty).\) It results by ([12]) that \(G/T\) \(\in FN_{k}^{(2)}\), therefore \(G\) is torsion-by-\(FN_{k}^{(2)}\), and this gives that \(G\) is \(\tau N_{k}^{(2)}\)-group.

Theorem 2.9. Let \(G\) a finitely generated \(\tau N\)-group. If \(G\) is in the class \(((\tau N_{k})\tau,\) \(\infty)\), then

1. \(G\) is \(\tau N_{k}^{(2)}\)-group.
2. There exist integers \(d\) such that \(G\) is in the class \(\tau N_{k^{d-1}}\).

Proof. (1) Assume that \(G\) is finitely generated \(\tau N\)-group in the class \(((\tau N_{k})\tau ,\infty)\). There exist a normal and torsion subgroup $H$ of \(G\) such that \(G/H\) is nilpotent quotient group. Since \(G/H\) is finitely generated nilpotent group, it has a torsion subgroup \(T/H\) of finite order and as \(H\) is torsion group then \(T\) is torsion group too. So \(G/T\) is torsion-free nilpotent group in the class \(((\tau N_{k})\tau ,\infty)\), which gives that \(G/T\) is in the class \((N_{k}\tau ,\infty)\). We deduce by ([18], Lemma 6.33) that \(G/T\) is in the class \((N_{k},\infty)\) and so \(G/T\) is a finitely generated soluble group in the class \((N_{k},\infty)\). It follows by [8] that \(G/T\) belongs in the class of \(FN_{k}^{(2)}\)-groups and torsion-free so \(G/T\) is in the class \(N_{k}^{(2)}\), it gives that \(G\) is in the class of \(\tau N_{k}^{(2)}\)-groups.
(2) In (1) we have \(G/T\) is a torsion-free nilpotent group in the class \(N_{k}^{(2)}\) which is included in \(\varepsilon _{k}\), so \(G/T\) is \(k\)-Engel torsion-free nilpotent (so soluble) group. If the integer \(d\) is the derived4length of \(G/T\) as a soluble group, then by a result of Gruenberg [18], Theorem 7.36, \(G/T\) is in the class \(N_{k^{d-1}}.\) So as \(T\) is torsion. It gives that \(G\) is \(\tau N_{k^{d-1}}\). This completes the proof.

If we replace the property \(\tau N\) by the property \(FN\), we obtain the results in the Lemma bellow.

Lemma 2.10. Let \(G\) a finitely generated \(FN\)-group in the class \(((FN_{k})F,\infty)\), then,

1. \(G\) is in the class of \(FN_{k}^{(2)}\)-groups.
2. There exist integers \(d =d(k)\) and \(c=c(k, d)\) such that \(G\) is in the class \(FN_{k^{d-1}}\) and \(G/Z_{c}(G)\) is finite.

Proof. (1) Assume that \(G\) is finitely generated \(FN\)-group in the class \(((FN_{k})F,\infty)\subset ((\tau N_{k})\tau ,\infty)\). As \(G\) is \(FN\)-group, there exist a normal and finite subgroup \(H\) of \(G\) such that \(G/H\) is nilpotent. We found that the torsion subgroup \(T/H\) of \(G/H\) is finite and so \(T\) is finite too. As the property \(((\tau N_{k})\tau,\infty)\) is closed under quotient then the quotient group \(G/T\) verifies the conditions of the above theorem. It follows that \(G/T\) belongs in the class of \(\tau N_{k}^{(2)}\)-groups which gives that \(G/T\) is in the class of \(N_{k}^{(2)}\)-groups and so G is \(FN_{k}^{(2)}\).
(2) In one hand as the same way in (2) of the above theorem we found that \(G/T\) is in the class \(N_{k^{d-1}}\) and \(T\) is finite. So \(G\) is in the class \(FN_{k^{d-1}}\). In the other hand and by Hall [15], there exist an integer \(c=c(k, d)\) depending on \(k, d\) such that \(G/Z_{c}(G)\).

The example 2.5 above shows that nilpotency is necessary for the results of the above theorem to remain true. Recall that \(FN\)-groups are \(NF\)-groups (see [15]).

Theorem 2.11. Let \(G\) a finitely generated \(NF\)-group. If \(G\) in the class \(((FN_{k})F, \infty)\), then

1. \(G\) is in the class of \(N_{k}^{(2)}F\)-groups. In particular, if \(G\) in the class \(((FA)F, \infty )\), then \(G\) is in the class of \(AF\)-groups.
2. There exist integers \(d=d(k)\) such that G is in the class \(N_{k^{d-1}}F\).

Proof. (1) Assume that \(G\) is an infinite finitely generated \(NF\)- group in the class \(((FN_{k})\tau , \infty )\). As the group \(G\) is \(NF\)-group, and then it contains a normal nilpotent subgroup \(N\) such that \(G/N\) is finite. As the subgroup \(N\) is finitely generated and nilpotent of finite index then \(N\) is polycyclic so by ([19], Theorem 5.4.15) there exist a subgroup \(M\) normal in \(N\) and poly-infinite cyclic, hence torsion-free and of finite index in \(N\). Let \(K=M_{G}\) the core of the subgroup \(M\), so \(K\) is nilpotent torsion-free of finite index in \(G\). Since the class \(((FN_{k})\tau,\infty)\) is closed under taking subgroups, then \(K\) is nilpotent subgroup in the class \(((FN_{k})\tau,\infty)\) and according to (1) of lemma 2.10, we deduce that \(K\) is torsion-free subgroup in the class of \(\tau N_{k}^{(2)}\)-groups which gives that \(K\) is \(N_{k}^{(2)}\)-group and so \(G\) is \(N_{k}^{(2)}F\)-group. In particular, for \(k=1\) we have: \(FN_{1})\tau =(FC)\tau=(FA)\tau\) and \(N_{1}^{(2)}\tau =A\tau\).
(2) As \(K\) is a torsion-free nilpotent subgroup in the class \(N_{k}^{(2)}\) then it is in the class \(\varepsilon_{k}\) of k-Engel groups. So by Gruenberg [[18], Theorem 7.36(1)], there exist integer \(d=d(k)\) such that \(K\) is in the class \(N_{k^{d-1}}\) and as \(K\) is of finite index then \(G\) is in the class of \(N_{k^{d-1}}F.\) This completes the proof.

In 2007 T.Rouabehi and N.Trabelsi in [17] proved that a finitely generated soluble group in the class \((\tau N_{k},\infty )^{\ast }\) is in the class \(\tau N_{c}\) for certain integer \(c\) depending only on \(k.\)
If we replace the properties \(((\tau N_{k})\tau , \infty )\) and \(((FN_{k})F,\infty )\) in the above results by the properties \(((\tau N_{k})\tau,\infty )^{\ast}\) and \(((FN_{k})F,\infty)^{\ast}\), we obtain the next results.

Theorem 2.12. Let \(G\) a finitely generated \(\tau N\)-group. \(G\) is in the class \(((\tau N_{k})\tau, \infty)^{\ast}\), then there exist an integer \(c=c(k)\) such that \(G\) is in the class of \(\tau N_{c}\)-group.

Proof. Assume that \(G\) is finitely generated \(\tau N\)- group in the class \(((\tau N_{k})\tau,\infty )^{\ast}\). There exist a normal and torsion subgroup \(F\) of \(G\) such that \(G/F\) is nilpotent quotient group. Since \(G/F\) is finitely generated nilpotent group, it has a finite and so torsion subgroup \(T/F\) such that \(T\) is a normal and torsion subgroup containing \(F\). So \(G/T\) is torsion-free nilpotent group in the class \(((\tau N_{k})\tau,\infty)^{\ast}\) and hence \(G/T\) is in the class \((N_{k}\tau,\infty)^{\ast }\). We deduce by ([18], Lemma 6.33) that \(G/T\) is in the class \((N_{k},\infty)^{\ast}\). It is known that the class \((N_{k},\infty )^{\ast }\) is included in the class \(\varepsilon _{k+1}(\infty)\), where \(\varepsilon_{k+1}(\infty)\) is the class of groups whose every infinite subset \(X\) contain two distinct elements \(x,\) \(y\) such that \([x,_{k+1}y]=1\). We deduce that \(G/T\) belongs in \(\varepsilon_{k+1}(\infty)\). Since \(G/T\) is nilpotent so soluble then by ([20], Theorem 3) there exist an integer \(c=c(k)\) depending only on \(k\) such that \((G/T)/Z_{c}(G/T)\) is finite. By a result in ([15], Theorem 1) \(\gamma _{c+1}(G/T)=\gamma _{c+1}(G)T/T\) is finite and so is torsion, and since \(T\) is torsion group , we deduce that \(\gamma _{c+1}(G)\) is torsion group. Therefore \(G\) is in the class of \(\tau N_{c}\)-group.
This completes the proof.

Lemma 2.13 Let \(G\) a finitely generated \(FN\)-group. Then

1. if \(G\) is in the class \(((FN_{k})F,\infty )^{\ast }\), then there exist an integer \(c=c(k)\) depending only on \(k\) such that \(G\) is in the class of \(FN_{c}\)-group.
2. if \(G\) is in the class \(((FC)F, \infty )^{\ast }\), then, \(G/Z_{2}(G)\) is finite and \(G\) is in the class of \(FN_{2}\)-groups.
3. if \(G\) is in the class \(((FN_{2})F, \infty )^{\ast }\), then, \(G\) is in the class of \(FN_{3}^{(2)}\)-groups and there exist an integer d such that \(G\) is \(FN_{3^{d-1}}\).

Proof. (1) Assume that \(G\) is finitely generated \(FN\)-group in the class \(((FN_{k})F, \infty)^{\ast}\). As \(G\) is \(FN\)-group, there exist a normal and finite subgroup \(F\) of \(G\) such that \(G/F\) is nilpotent quotient group. Since \(G/F\) finitely generated nilpotent group it has a torsion subgroup \(T/F\) of finite order. So the subgroup \(T\) of torsion elements of \(G\) is normal and finite in \(G\) and as the same way in the above theorem, we deduce by ([18], Lemma 6.33) that \(G/T\) is nilpotent torsion -free in the class \((N_{k},\infty )^{\ast }\subset \varepsilon_{k+1}(\infty)\) and according to ([20], Theorem 3) we found that there exist an integer \(c=c(k)\) depending only on \(k\) such that \((G/T)/Z_{c}(G/T)\) is finite. Also by ([15], Theorem 1) we find that \(\gamma _{c+1}(G)T/T\) is finite and as \(T\) is finite then \(\gamma _{c+1}(G)\) is finite too. Therefore \(G\) is in the class of \(FN_{c}\)-group.
(2) If \(G\) is in the class \(((FC)F,\infty )^{\ast }=((FA)F,\infty )^{\ast}=((FN_{1})F, \infty )^{\ast }\), then as in (i) \(G/T\) is in the class \((N_{1},\infty )^{\ast }\subset \varepsilon _{2}(\infty),\) by a result of Abdollahi [11] \((G/T)/Z_{2}(G/T)\) is finite and by ([15], Theorem 1) we find that \(\gamma _{3}(G)T/T\) is finite and as \(T\) is finite then \(\gamma _{3}(G)\) is finite too and as \(G\) is finitely generated then by [15],\(G/Z_{2}\) is finite. Therefore \(G\) is in the class of \(FN_{2}\)-group.
(3) For \(k=2\), as the same way in (1) we found that \(G/T\) is in the class \((N_{2},\infty )^{\ast }\) which is included in the class \(\varepsilon{3}(\infty )\), where \(\varepsilon_{3}(\infty )\) is the class of groups whose every infinite subset \(X\) contain two distinct elements \(x, y\) such that \([x,_{3}y]=1\). We deduce by (20, Theorem 1) that \(G/T\) is torsion-free in the class \(FN_{3}^{(2)}\) so G/T is in \(N_{3}^{(2)}\) and as the torsion subgroup \(T\) is finite, then \(G\) is \(FN_{3}^{(2)}\)-group. As \(G/T \) is torsion-free soluble group in \(N_{3}^{(2)}\subset \varepsilon _{3}\) (the 3-Engel group) then by Gruenberg [[18], Theorem 7.36 (1)], there exist integer d such that \(G/T\) is in the class \(N_{3^{d-1}}\) which gives that \(G\) is \(FN_{3^{d-1}}\). This completes the proof.

Theorem 2.14. Let \(G\) a finitely generated \(NF\)-group. Then

1. if \(G\) is in the class \(((FN_{k})F,\) \(\infty )^{\ast }\), then there exist an integer \(c=c(k)\) depending only on such that \(G\) is in the class of \(N_{c}F\)-groups.
2. if \(G\) is in the class of \(((FC)F, \infty )^{\ast }\)-groups, then, \(G\) is in the class of \(N_{2}F\)-group.
3. if \(G\) is in the class \(((FN_{2})F, \infty )^{\ast }\), then, \(G\) is in the class of \(N_{3}^{(2)}F\)-groups and there exist an integer \(d\) such that \(G\) is \(FN_{3^{d-1}}\).

Proof. (1) As the group \(G\) is \(NF\)- group, and then it contains a normal nilpotent subgroup \(N\) such that \(G/N\) is finite. As the subgroup \(N\) is finitely generated and nilpotent of finite index then \(N\) is polycyclic so by ([19], Theorem 5.4.15) there exist a normal subgroup \(M\) in \(N\) and poly-infinite cyclic hence torsion-free and of finite index in \(N\). Let \(K=M_{G}\) the core of the subgroup \(M\), so \(K\) is nilpotent torsion-free of finite index in \(G.\) Since the class \(((FN_{k})F, \infty )^{\ast }\) is closed under taking subgroups, then \(K\) is in this class too, so by (1) of lemma 2.13, we obtains that there exist an integer \(c=c(k)\) depending only on \(k\) such that \(K\) is \(FN_{c}\)-group and as \(K\) is torsion-free, it is \(N_{c}\)-group and so \(G\) is \(N_{c}F\)-group.
(2) Particularly for \(k=1\), we have \(((FC)F, \infty )^{\ast }=((FN_{1})F,\infty )^{\ast }\), in this case the subgroup \(K\) is a finitely generated torsion-free nilpotent group in the class \(((FN_{1})F, \infty )^{\ast }\) and according to (2) of lemma 2.13, we deduce that \(K\) is in the class \(FN_{2}\)-groups and as \(K\) is torsion-free, it is \(N_{2}\)-group of finite index in \(G\), this gives that \(G\) is \(N_{2}F\)-group.
(3) In particular for \(k=2\), we have the subgroup \(K\) in (1) is a finitely generated torsion-free nilpotent group in the class \(((FN_{2})F,\infty )^{\ast }\) and according to (3) of lemma 2.13, we deduce that \(K\) is in the class \(FN_{3}^{(2)}\)-groups and as \(K\) is torsion-free it is the class \(N_{3}^{(2)}\)-group and as \(G/K\) if finite this gives that \(G\) is in the class of \(N_{3}^{(2)}F\)-groups. As K is nilpotent torsion- free in the class \(N_{3}^{(2)}\) then it is in the class \(\varepsilon _{3}\) of 3-Enjel group, then by Gruenberg [18, Theorem 7.36 (1)], there exist integer \(d\) such that \(K\) is in the class \(N_{3^{d-1}}\) and so \(G\) is in the class \(N_{3^{d-1}}F\)-group.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions in advance.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Let \(G\) be a simple connected graph with vertex set \(V(G)\) and edge set \(E(G)\). Let \(n\) and \(m\) respectively denote the order and size of \(G\). In this paper we consider only simple connected graph, that is, graphs without any self loop or parallel edges. In molecular graph theory, molecular graphs represent the chemical structures of a chemical compound and it is often found that there is a correlation between the molecular structure descriptor with different physico-chemical properties of the corresponding chemical compounds. These molecular structure descriptors are commonly known as topological indices which are some numeric parameter obtained from the molecular graphs and are necessarily invariant under automorphism. Thus topological indices are very important useful tool to discriminate isomers and also shown its applicability in quantitative structure-activity relationship (QSAR), structure-property relationship (QSPR) and nanotechnology including discovery and design of new drugs [1, 2].

Among different types of topological indices, the vertex degree based topological indices are very useful topological indices in study of structure property correlation of molecules. The degree of a vertex \(v\) of a graph \(G\) is equal to number of vertices adjacent with it and is denoted by \(d(v)\) whereas \(\delta (v)\) represents the sum of degrees of all the vertices adjacent to \(v\). The first and second Zagreb indices are the oldest vertex degree based topological indices and was introduced by Gutman and Trinajstić [3] and are defined as \[{{M}_{1}}(G)=\sum\limits_{v\in V(G)}{{{d}_{G}}{{(v)}^{2}}}=\sum\limits_{uv\in E(G)}{[{{d}_{G}}(u)+{{d}_{G}}(v)]}\] and \[{{M}_{2}}(G)=\sum\limits_{uv\in E(G)}{{{d}_{G}}(u){{d}_{G}}(v)}.\] These indices have been extensively studied both with respect to mathematical and chemical point of view. There are various extension of Zagreb indices. In [4], Miličević et al. Introduced a new version of Zagreb indices called reformulated Zagreb index. Here the degree of a vertex of classical Zagreb indices is replaced by degree of an edge \(e\) denoted by \(d(e)\), where the degree of an edge \(e=uv\) is given by \(d(u)+d(v)-2\). Therefore, the first and second reformulated Zagreb indices of a graph \(G\) are defined as \[{E}{{M}_{1}}(G)=\sum\limits_{e\in E(G)}{{d}{{(e)^2}}}=\sum\limits_{e=uv\in E(G)}{[{{d}_{G}}(u)+{{d}_{G}}(v)-{2}]^2},\] \[{E}{{M}_{2}}(G)=\sum\limits_{e \sim f\in E(G)}{{d}{{(e)}}{{d}{{(f)}}}}.\] In the definition of second reformulated Zagreb index, the notation \(e\sim f\) means that the edges \(e\) and \(f\) are adjacent, that is, they share a common vertex in \(G\). There are a large number of studies regarding various chemical and mathematical properties of the reformulated Zagreb indices. Different basic properties and bounds of reformulated Zagreb indices have been studied in [5] and [6]. In [7], bounds for the reformulated first Zagreb index of graphs with connectivity at most \(k\) are obtained. De [8], found some upper and lower bounds of these indices in terms of some other graph invariants and also derived reformulated Zagreb indices of a class of dendrimers in [9]. Ji et al. in [10] and [11], computed these indices for acyclic, unicyclic, bicyclic and tricyclic graphs. Recently De et al. [12], investigate reformulated first Zagreb index of various graph operations.

We know that, different graph operations create a new graph from the given simpler graphs and sometimes it is found that, some chemically interesting graphs can be obtained as a result of graph operations of some simpler graphs. So from the relations obtained for different topological indices of graph operations in terms of topological indices of their components, it is convenient to find topological indices of some special molecular graphs and nanostructures. There are several study concerning topological indices of different graph operations. In [13], [14] and [15], De et al. obtained connective eccentric index, F-index and F-coindex of different graph operations respectively. Khalifeh et al., in [16] obtained some exact expressions for computing first and second Zagreb indices of different graph operations. In [18], Ashrafi et al. compute expressions for Zagreb coindices of different graph operations. In [18], Das et al. found some upper bounds for multiplicative Zagreb indices of some graph operations and Azari et al. in [19], derived explicit formulae for computing the eccentric-distance sum of different graph operations. Recently, De computes the vertex Zagreb indices of different graph operations in [20]. For more results, interested readers are referred to the papers [21, 22, 23, 24]. In this study, our goal is to study reformulated first Zagreb index of an important graph operation called generalized hierarchical product of graphs to obtain this topological index of different chemical graphs. Barrire et al. in 2009 introduced the generalized hierarchical product of graphs [25],which is a generalization of both Cartesian product of graphs and the (standard) hierarchical product of graphs [26]. There are various studies of generalized hierarchical product of graphs for different topological indices in recent literature [27, 28, 29, 30, 31].

In this paper, we first derive explicit expression of reformulated first Zagreb index of generalized hierarchical product of two connected graphs. Hence using the derived results, the reformulated first Zagreb index of some chemically important graphs such as square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer fullerene etc. are obtained.

2. Generalized Hierarchical Product of Graphs

The generalized hierarchical product of graphs is one of most important graph operations as many other graph operations such as Cartesian product of graphs, cluster product of graphs can be considered as a special case of this graph operation. There are already various studies on different topological indices of this graph operation till date.

Definition 2.1. Let \(G\) and \(H\) be two connected graphs and \(U\) be a non-empty subset of \(V(H)\). Then the hierarchical product of \(G\) and \(H\), denoted by \(G\sqcap H(U)\), is the graph with vertex set \(V(G)\times V(H)\), and any two vertices \((u,v)\) and \(u^{\prime},v^{\prime}\) of \(G\sqcap H(U)\) are adjoint by an edge if and only if \([v=v^{\prime} \;\; \text{and} \;\; uu^{\prime}\in E(G)]\) or \([v=v^{\prime}\;\; \text{and} \;\; vv^{\prime}\in E(H)].\)

From definition, we can state the following lemma which gives some basic properties of generalized hierarchical product of graphs.

Lemma 2.2. Let \(G\) and \(H\) be graphs with \(U\subseteq V(H).\) Then

1. \(G\sqcap H(U)\) is connected if and only if \(G\) and \(H\) are connected.
2. The degree of a vertex \((a,x)\) of \(G\sqcap H(U)\) is given by \[{{d}_{{{G}_{U}}\sqcap{H}}}(a,x)= \left\{ \begin{array}{ll} {d}(x)+{χ}_{U}(x)d(a),~a\in V({{G}}),~x\in E(H)\\[2mm] d(x)+d(a),~a\in V(G),~x\in U\\[2mm] \end{array}\right.\]

where \(χ_{U}(x)\) is characteristic function on the set \(U,\) which is 1 on \(U\) and 0 outside \(U.\)

Now let us first derive the reformulated first Zagreb index of generalized hierarchical product of graphs \(G\) and \(H\).

Theorem 2.3. The reformulated first Zagreb index of \(G\sqcap H(U)\) is given by \begin{eqnarray*} EM_1(G\sqcap H(U))&=&|V(G)|EM_1(H)+|U|EM_1(G)+5M_1(G)\sum\limits_{u\in U}d(u)\\ &&+8|E(G)|\sum\limits_{u\in U}d(u)^2+2M_1(G)\{xy\in E(H): x,~y\in U\}\\ &&+4|E(G)|\sum\limits_{u\in U}\sum\limits_{x\in N[u]}d(x)-16|E(G)|\sum\limits_{u\in U}d(u). \end{eqnarray*}

Proof. Let \(G\) and \(H\) be two connected graphs and \(U\) be a nonempty subset of \(V(H)\). Let us partition the edge set of \(G\sqcap H(U)\) into two subsets, say \(E_1\) and \(E_2\) so that \[E_1=\{(a,x)(b,y): xy\in E(H)\;\; and\;\; a=b\in V(G)\}\] \[E_2=\{(a,x)(b,y): xy\in E(G)\;\; and\;\; x=y\in V(H)\}.\] Now using definition of reformulated first Zagreb index and generalized hierarchical product graph, we consider the following two cases to calculate the reformulated first Zagreb index of \(G\sqcap H(U)\).
Case 1. The contributions of the edges of \(H_1\) to the reformulated first Zagreb index of \(G\sqcap H(U)\) is calculated as \begin{eqnarray*} H_1&=&\sum\limits_{a\in V(G),xy\in E(H)}\{d(a,x)+d(a,y)-2\}^2\\ &=&\sum\limits_{a\in V(G),xy\in E(H)}\{d(x)+χ_{U}(x)d(a)+d(y)+χ_{U}(y)d(a)-2\}^2\\ &=&\sum\limits_{a\in V(G),xy\in E(H)}\{d(x)+d(y)-2\}^2\\&&+\sum\limits_{a\in V(G),xy\in E(H)}\{χ_{U}(x)d(a)+χ_{U}(y)d(a)\}^2\\ &&+2\sum\limits_{a\in V(G),xy\in E(H)}d(a)\{d(x)+d(y)-2\}\{χ_{U}(x)+χ_{U}(y)\}\\ &=&|V(G)|EM_1(H)+\sum\limits_{a\in V(G),xy\in E(H)}d(a)^2\{χ_{U}(x)^2+χ_{U}(y)^2\\ &&+2χ_{U}(x)χ_{U}(y)\}+2\sum\limits_{a\in V(G),xy\in E(H)}d(a)\{(χ_{U}(x)d(x)+χ_{U}(y)d(y))\\ &&+(χ_{U}(x)d(y)+χ_{U}(y)d(x))-2(χ_{U}(x)+χ_{U}(y))\}\\ &=&|V(G)|EM_1(H)+\sum\limits_{a\in V(G),xy\in E(H)}d(a)^2\{χ_{U}(x)^2+χ_{U}(y)^2\}\\ &&+2\sum\limits_{a\in V(G),xy\in E(H)}d(a)^2χ_{U}(x)χ_{U}(y)\\ &&+2\sum\limits_{a\in V(G),xy\in E(H)}d(a)\{χ_{U}(x)d(x)+χ_{U}(y)d(y)\}\\ &&+2\sum\limits_{a\in V(G),xy\in E(H)}d(a)\{χ_{U}(x)d(y)+χ_{U}(y)d(x)\}\\ &&-4\sum\limits_{a\in V(G),xy\in E(H)}d(a)\{χ_{U}(x)+χ_{U}(y)\}\\ &=&|V(G)|EM_1(H)+M_1(G)\sum\limits_{u\in U}d(u)+2M_1(G)|\{xy\in E(G) : x,y \in U\}|\\ &&+4|E(G)|\sum\limits_{u\in U}d(u)^2+4|E(G)|\sum\limits_{u\in U}\sum\limits_{x\in N[u]}d(x)-8|E(G)|\sum\limits_{u\in U}d(u). \end{eqnarray*} Case 2. Similarly, the contributions of the edges of \(H_2\) to the reformulated first Zagreb index of \(G\sqcap H(U)\) is calculated as. \begin{eqnarray*} H_2&=&\sum\limits_{ab\in E(G),x\in U}\{d(a,x)+d(b,x)-2\}^2\\ &=&\sum\limits_{ab\in E(G),x\in U}\{d(x)+d(a)+d(x)+d(b)-2\}^2\\ &=&\sum\limits_{ab\in E(G),x\in U}\{2d(x)+(d(a)+d(b)-2)\}^2\\ &=&4\sum\limits_{ab\in E(G),x\in U}d(x)^2+\sum\limits_{ab\in E(G),x\in U}\{d(a)+d(b)-2\}^2\\ &&+4\sum\limits_{ab\in E(G),x\in U}d(x)\{d(a)+d(b)\}-8\sum\limits_{ab\in E(G),x\in U}d(x). \end{eqnarray*} Finally, combining the contributions of \(H_1\) and \(H_2\), the desired result follows.

Now we consider reformulated first Zagreb index of a particular generalized hierarchical product graph, where \(G=P_2\), as some chemical structures are in this form.

Corollary 2.4. The reformulated first Zagreb index of cluster product of two graphs \(G\) and \(H\), with root vertex of degree one is given by \begin{eqnarray*} EM_1(P_2\sqcap H(U))&=&2EM_1(H)+8\sum\limits_{u\in U}d(u)^2-6\sum\limits_{u\in U}d(u)\\ &&+4|\{xy\in E(H): x,y\in U\}|+4\sum\limits_{u\in U}\sum\limits_{x\in N[u]}d(x). \end{eqnarray*} The Cartesian product of two graphs \(G\) and \(H\) denoted by \(G\times H\) is the graph with the vertex set \(V(G)\times(H)\), with vertices \(u=(u_1,u_2)\) and \(v=(v_1,v_2)\) connected by an edge if and only if \([u_1=v_1\; and\; (u_2,v_2)\in E(G_2)]\) or \([u_2=v_2\;\; and\;\; (u_1,v_1)\in E(G_1)].\) From definition it is clear that the generalized hierarchical product of graph is a subset of cartesian product of graphs. Also, if \(U=V(H)\), then \(G\sqcap H(U)=G\times H.\) Thus from Theorem 2.3 the following result follows:

Theorem 2.5. The reformulated first Zagreb index of Cartesian product of two graphs \(G\) and \(H\) is given by \begin{eqnarray*} EM_1(G\times H)&=&|V(G)|EM_1(H)+|V(H)|EM_1(G)+12|E(G)|M_1(H)\\ &&+12|E(H)|M_1(G)-32|E(G)||E(H)|. \end{eqnarray*}

Note that, the exact formula for reformulated first Zagreb index of the Cartesian product of graphs was obtained directly in [12], which coincides with the above result. For different special graphs obtained by specializing components of the cartesian product of graphs such as \(C_4\) nanotube \(P_n\times C_m,\) \(C_4\) nanotorus \(C_n\times C_m\), n-prism \(K_2\times C_n\), ladder graph \(P_2\times P_{n+1}\), grid graph \(P_n\times P_m\) rooks graph \(K_n\times K_m\) are discussed there in.

2.1. The cluster product of graphs

The cluster product of two graphs \(G\) and \(H\) denoted by \(G\{H\}\) is the graph obtained by taking one copy of \(G\) and \(V(G)\) copies of a rooted graph \(H\), and by identifying the root of the i-th copy of \(H\) with the i-th vertex of \(G,\) for \(i=1,2,....|V(G)|.\)

The cluster of two graphs \(G\) and \(H\) with root vertex \(x\), can be represented as a special case of generalized hierarchical product of graphs if \(U=\{x\}\), a singleton. Thus we have, \(G\sqcap H(U)=G\{H\}\). The cluster product of two graphs \(G\) and \(H\) with root \(x\) is also termed as (standard) hierarchical product and is denoted by \(G\sqcap (H=G\{H\})\). Thus we have, \(G\sqcap H(U)=G\{H\}=G\sqcap H\), if \(U=\{x\}\). Thus from Theorem 2.3, by considering \(U=\{x\}\), we get the following result directly.

Theorem 2.6. The reformulated first Zagreb index of cluster product of two graphs \(G\) and \(H\), with root vertex \(x\) of \(H\) is given by \begin{eqnarray*} EM_1(G\{H\})&=&|V(G)|EM_1(H)+EM_1(G)+5d(x)M_1(G)+8|E(G)|d(x)^2\\ &&+4|E(G)|\delta (x)-16|E(G)|d(x). \end{eqnarray*}

Let \(G=(V(G),E(G))\) be a \(r\)-regular graph, then \(d(x)=r,\) \(\delta (x)=r^2,\) \(M_1(G)=r^2|V(G)|\) and \(HM(G)=(2r-2)^2|E(G)|=2r(r-1)^2|V(G)|.\) Then the following result follows directly from above Theorem 2.6.

Corollary 2.7. If \(G\) and \(H\) be \(r\) and \(s\)-regular graph, respectively, then \[EM_1(G\{H\})=|V(G)|\{2s(s-1)^2|V(H)|+2r(r-1)^2+5r^2s+6rs^2-8rs\}.\]

Using the above result, the following example follows:

Example 2.8. \begin{eqnarray*} (i)~EM_1(C_n\{C_m\})&=&4n(m+15)\\ (ii)~EM_1(K_n\{K_m\})&=&n[2m(n-1)(n-2)^2+2(m-1)(m-2)^2\\ &&+5(n-1)^2(m-1)+6(n-1)(m-1)^2\\ &&-8(n-1)(m-1)]. \end{eqnarray*}

The \(t\)-thorny graph or the \(t\)-fold bristled graph of a graph \(G\) is obtained by joining \(t\) number of degree one vertices or thorn to each vertex of \(G\). This type of graph was introduced by Gutman [32] and studied by different authors also [33, 34, 35]. From construction it is clear that the \(t\)-thorny graph is the cluster product of \(G\) and a star graph \(S_{t+1}\), where the root vertex is the central vertex of \(S_{t+1}\) with degree \(t\). Hence, using Theorem 2.6 we get the following result.

Oorollary 2.9. Let \(G\) be a connected graph with \(n\) vertices and \(m\) edges, then \[EM_1(G\{S_{t+1}\})=EM_1(G)+5tM_1(G)+nt(t-1)^2+8mt^2-12mt.\]

We have derived the above result in \cite{10} also as a special case of corona product of graphs and hence different results for some particular thorn graphs were also derived there. Let us now consider one important type of graph which is the cluster product of \(G\) and \(P_2\). Using Theorem 2.6, its first reformulated Zagreb index is calculated as follows:

Corollary 2.10. Let \(G\) be a connected graph with \(n\) vertices and \(m\) edges, then \[EM_1(G\{P_2\})=EM_1(G)+5M_1(G)-4|E(G)|.\]

3. Application of Generalized Hierarchical Product and Cluster Product of Graphs

We know that, many chemically interesting molecular graphs and nanostructures are obtained from different graph operations of some particular graphs, so in this section we investigate such particular molecular graphs which are resultant of generalized hierarchical product and cluster product of some particular graphs. Thus, we apply the results derived in the previous section to compute reformulated first Zagreb index of some special chemically interesting molecular graphs and also of nanostructures. First we consider some chemical structure in the form \(P_2\sqcap H(U).\)

Example 3.1. The molecular graph of truncated cube can be considered as generalized hierarchical product of \(P_2\) and \(H(U)\) (shown in Figure 1) , where \(U=\{v_1,v_4,v_9,v_{12}\}\) . Since, \(EM_1(H)=2000\), thus using Corollary 2.4, we get \(EM_1(P_2\sqcap H(U))=576.\)

Example 3.2. The dimer fullerene is the generalized hierarchical product of \(P_2\) and \(C_{60}\), where \(U=\{5,6\}\) ( as shown in Figure 2). From direct calculation, we have \(EM_1(C_{60})=1440\). Thus, using Corollary 2.4, we get \(EM_1(P_2\sqcap C_{60}(U))=3064.\)

Example 3.3. Let \(L_n\) be the hexagonal chain with \(n\) hexagons (shown in Figure 3). From definition it is clear that \(L_n=P_2\sqcap P_{2n+1}(U)\), where \(U=\{v_1,v_3,v_5,.......,v_{2n+1}\}\). Then using Corollary 2.9, we get \[EM_1(L_n)=EM_1(P_2\sqcap P_{2n+1}(U))=52n-28.\]

Example 3.4. The zig-zag polyhex nanotube \(TUHC_6[2n,2]\) is the generalized hierarchical product of \(P_2\) and \(C_{2n}(U)\), where \(U=\{v_2,v_4,........,v_{2n}\) (Figure 4). Thus using Corollary 2.4, we get \[EM_1(TUHC_6[2n,2])=EM_1(P_2\sqcap C_{2n}(U))=52n.\]

Example 3.5. The linear phenylene \(F_n\) with \(n\) benzene ring is the graph given by \(P_2\sqcap P_{3n}(U)\), where \(U=\{v_{3k+1}: 0\leq k \leq n-1\}\cup \{v_{3k}:1\leq k \leq n.\}\) Then using Corollary 2.4, we get the following result \[EM_1(F_n)=EM_1(P_2\sqcap P_{3n}(U))=96n-72.\]

In the following example, now we consider some important molecular structures which can be obtained as a result of cluster product of some special graphs. First let us consider one important nanostructure called regular dicentric dendrimers.

Example 3.6. Let \(DD_{p,r}\) denote the regular dicentric dendrimers. Then \(DD_{p,r}=P_2\{H\}\) , where \(H\) is a dendron tree of progressive degree \(p\) and generation \(r\) (see Figure 6). Thus using theorem 2.6, we get \[EM_1(DD_{p,r})=2EM_1(H)+12p^2-2p.\] Since from direct calculation, we have \[EM_1(H)=\frac{p(p^{r+2}+3p^{r+1}-8p^2+5p-1)}{p-1}.\] So, we get from above \[EM_1(DD_{p,r})=\frac{2p^2(p^{r+1}-3p^r-2p-2)}{p-1}.\]

Example 3.8. The sun graph denoted by \(Sun_{(m,n)}\) is defined as cluster product of \(C_m\) and \(P_{n+1}\), where \(P_{n+1}\) is rooted at a vertex of degree one. Then using Theorem 2.6, the reformulated first Zagreb index of \(Sun_{(m,n)}\) is given by \[EM_1(Sun_{(m,n)})=2m(2n+9).\]

Example 3.9. Let \(\Gamma\) be the molecular graph of octanitrocubane (see Figure 7). It can be considered as the cluster product of a cube \((G)\) and \(P_2\). Then using Corollary 2.4, we similarly get \(EM_1(\Gamma)=504.\)

Example 3.10. The square comb lattice graph \(C_q(N)\) with open ends, can be considered as the cluster product \(P_n\{P_n\},\) where \(N=n^2\) is the total number of vertices of \(C_q(N)\) (see Figure 8). Here root of \(P_n\) is on its pendent vertex i.e. the vertex of degree one. Thus applying Theorem 2.6 we get, for \(n\ge 3\) \[EM_1(C_q(N))=n(8n-14)+6(n-1)+6(2n-3)+6n-14=8n^2+10n-38.\]

4. Conclusion

In this paper, we first derive explicit expression of reformulated first Zagreb index of generalized hierarchical product of two connected graphs and hence using this result we obtain the reformulated first Zagreb index of cluster product of two graphs. Finally, the reformulated first Zagreb index of some molecular graphs such as square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer fullerene etc. are obtained.

Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

In early years, many chemical experiments showed the evidence that the biochemical properties of chemical compounds, materials and drugs are closely related to their molecular structures. As a result, topological indices are introduced as numerical parameters of molecular graph, which play a vital role in understanding the properties of chemical compounds and are applied in disciplines such as chemistry, physics and medicine science.

In chemical graph theory, a molecular structure is expressed as a molecular graph \(G\) in which atoms are taken as vertices and chemical bonds are taken as edges. A topological index can be considered as a function \(f:G\rightarrow \mathcal{R}^{+}\). In the past 40 years, scholars introduced many topological indices, such as Wiener index, Zagreb index, harmonic index, sum connectivity index, etc which reflect certain structural characteristics of organic molecules. There were many works contributing to report these distance-based or degree-based indices of special molecular structures (See Farahani et al. [1], Jamil et al. [2], Gao and Farahani [3], Gao et al. [4, 5, 6] and Gao and Wang [7, 8, 9] for details). The notation and terminology that were used but were undefined in this paper can be found in [10].

One of oldest indices, the Wiener index was defined as the sum of distance for all pair of vertices, $$W(G)=\sum_{\{u,v\}\subseteq V(G)}d(u,v).$$ The modified Wiener index was introduced by Nikolić et al. [11] as the extension of the Wiener index which was defined as $$W_{\lambda}(G)=\sum_{\{u,v\}\subseteq V(G)}d^{\lambda}(u,v).$$ Several conclusions on modified Wiener index can be referred to Vukičević and Žerovnik [12], Vukičević and Gutman [13], Lim [14], Gorse and Žerovnik [15], Vukičević and Graovac [16], and Gutman et al. [17].

Moreover, the hyper-Wiener index and \(\lambda\)-modified hyper-Wiener index are defined as $$WW(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{2}(d(u,v)+d^{2}(u,v))$$ and $$WW_{\lambda}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{2}(d^{\lambda}(u,v)+d^{2\lambda}(u,v)),$$ respectively. Some important contributions on hyper-Wiener index can be found in Gutman [18], Gutman and Furtula [19], Eliasi and Taeri [20, 21], Iranmanesh et al. [22], Yazdani and Bahrami [23], Behtoei et al. [24], Mansour and Schork [25], Heydari [26], Ashrafi et al. [27], and Heydari [28].

The Harary index was introduced independently by Plavšić et al. [29] and Ivanciuc et al. [30] in 1993, as $$H(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)}.$$ Its corresponding Harary polynomial can be defined as $$H(G,x)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)}x^{d(u,v)}.$$ The second and third Harary indices are defined as $$H_{1}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)+1},$$ $$H_{2}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)+2}.$$ More generally, the generalized Harary index was introduced by Das et al. [31] which is defined as $$H_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{d(u,v)+t},$$ where \(t\in\mathcal{N}\) is a non-negative integer. Hence, Harary index is a special case of generalized Harary index when \(t=0\).

One topological index related to Wiener index is the reciprocal complementary Wiener (RCW) index which is defined by Zhou et al. [32] and can be defined as $$RCW(G)=\sum_{\{u,v\}\subseteq V(G)}\frac{1}{D(G)+1-d(u,v)},$$ where \(D(G)\) is the diameter of molecular graph \(G\). In what follows, we always denote \(D(G)\) as the diameter of molecular graph \(G\).

Furthermore, the multiplicative version of the Wiener index was defined by Gutman et al. [33, 34] as $$\pi(G)=\prod_{\{u,v\}\subseteq V(G)}d(u,v).$$ The logarithm of multiplicative Wiener index was defined as $$\Pi(G)=\ln\sqrt{2\prod_{\{u,v\}\subseteq V(G)}d(u,v)}.$$

So far, many mathematical approaches are given to calculate different topological indices and received good results. Since most of the chemical compounds have symmetric structures. It inspires us to consider the computation of topological indices by using group theory. We use the automorphism groups and its orbits to simplify the computation of molecular graphs for some distance-based indices.

2. Main results and proofs

To discuss the symmetry molecular structures, we should first introduce symmetry operations which are defined as operations that move a fixed molecule structure from a previous condition to another, and any two states can't be differentiated from each other. Obviously, all the symmetry operations on a molecular structure constitute a group which is called the point group of the molecular structure.

When an element \(p\) of point group \(P\) (i.e., a symmetry operation on the molecular structure) operates on the molecular graph, it provides the vertices of the molecular graph a permutation. We denote \(p(v)\) as the image of vertex \(v\) under the operation \(p\). If there exists a \(p\in P\) that satisfies \(p(v)=u\) for two vertices \(v\) and \(u\), then we define an equivalence binary relation denoted by \(v\sim u\). By means of this equivalent relation, the vertex set is divided into several equivalence classes: \(\Theta_{1},\cdots,\Theta_{r}\). Each \(\Theta_{i}\) can be called an orbit of point group \(P\), and the number of vertices of each orbit \(|\Theta_{i}|\) is called the length of the orbit \(\Theta_{i}\). By the knowledge of group theory, we know that if \(v\in \Theta_{i}\) then \(|\Theta_{i}|=\frac{|P|}{|P_{v}|}\), where \(P_{v}=\{p|p\in P,p(v)=v\}\). The group called transitive if it has only one orbit, and it is called intransitive otherwise. Moreover, we can define the orbits of subgroup \(H\) in similar way which could also be either transitive or intransitive.

Set

1. \(W_{\lambda}(v,G)=\sum_{u\in V(G)}d^{\lambda}(u,v),\)
2. \(WW(v,G)=\frac{1}{2}\sum_{u\in V(G)}(d(u,v)+d^{2}(u,v)),\)
3. \(WW_{\lambda}(v,G)=\frac{1}{2}\sum_{u\in V(G)}(d^{\lambda}(u,v)+d^{2\lambda}(u,v)),\)
4. \(H(v,G)=\sum_{u\in V(G)}\frac{1}{d(u,v)},\)
5. \(H(v,G,x)=\sum_{u\in V(G)}\frac{1}{d(u,v)}x^{d(u,v)},\)
6. \(H_{1}(v,G)=\sum_{u\in V(G)}\frac{1}{d(u,v)+1},\)
7. \(H_{2}(v,G)=\sum_{u\in V(G)}\frac{1}{d(u,v)+2},\)
8. \(H_{t}(v,G)=\sum_{u\in V(G)}\frac{1}{d(u,v)+t},\)
9. \(RCW(v,G)=\sum_{u\in V(G)}\frac{1}{D(G)+1-d(u,v)},\)
10. \(\pi(v,G)=\prod_{u\in V(G)}d(u,v).\)

Hence, we have

1. \(W_{\lambda}(G)=\frac{1}{2}\sum_{v\in V(G)}W_{\lambda}(v,G) \;\; \;\;\;\;\;\;\;\;\;\; \;\; \;\;\;\;\;\;\;\;\;\; \;\; \;\;\;\;\;\;\;\;\;\; \;\; \;\;\;\;\;\;\;\;\;\; \;\; \;\;\;\;\;\;\;\;\;\; \;\; \;\;\;\;\;\;\;\;\;\; (1)\)
2. \(WW(G)=\frac{1}{2}\sum_{v\in V(G)}WW(v,G),\)
3. \(WW_{\lambda}(G)=\frac{1}{2}\sum_{v\in V(G)}WW_{\lambda}(v,G),\)
4. \(H(G)=\frac{1}{2}\sum_{v\in V(G)}H(v,G),\)
5. \(H(G,x)=\frac{1}{2}\sum_{v\in V(G)}H(v,G,x),\)
6. \(H_{1}(G)=\frac{1}{2}\sum_{v\in V(G)}H_{1}(v,G),\)
7. \(H_{2}(G)=\frac{1}{2}\sum_{v\in V(G)}H_{2}(v,G),\)
8. \(H_{t}(G)=\frac{1}{2}\sum_{v\in V(G)}H_{t}(v,G),\)
9. \(RCW(G)=\frac{1}{2}\sum_{v\in V(G)}RCW(v,G),\)
10. \(\pi(G)=\sqrt{\prod_{v\in V(G)}\pi(v,G)},\)
11. \(\Pi(G)=\ln\sqrt{2\sqrt{\prod_{v\in V(G)}\pi(v,G)}}.\)

Following theorem is about the calculation of different topological indices when the point group is not necessarily transitive.

Theroem 2.1. Let \(H\unlhd P_{G}\) be a subgroup of \(P_{G}\), and \(\Theta_{1}, \Theta_{2},\cdots, \Theta_{r}\) are the orbits of \(H\) and \(u_{i}\in\Theta_{i}\), \(i=1,2,\cdots,r\). Then, we have

1. \(W_{\lambda}(G)=\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}d^{\lambda}(u,u_{i})+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in \Theta_{i}}d^{\lambda}(v,\\u_{i}),\)
2. \(WW(G)=\frac{1}{2}\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}(d(u,u_{i})+d^{2}(u,u_{i}))+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\\\sum_{v\in \Theta_{i}}(d(v,u_{i})+d^{2}(v,u_{i})),\)
3. \(WW_{\lambda}(G)=\frac{1}{2}\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}(d^{\lambda}(u,u_{i})+d^{2\lambda}(u,u_{i}))+\frac{1}{2}\sum_{i=1}^{r}\\| \Theta_{i}|\sum_{v\in \Theta_{i}}(d^{\lambda}(v,u_{i})+d^{2\lambda}(v,u_{i})),\)
4. \(H(G)=\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}\frac{1}{d(u,u_{i})}+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in \Theta_{i}}\frac{1}{d(v,u_{i})},\)
5. \(H(G,x)=\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}\frac{1}{d(u,u_{i})}x^{d(u,u_{i})}+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in \Theta_{i}}\\ \frac{1}{d(v,u_{i})}x^{d(v,u_{i})},\)
6. \(H_{t}(G)=\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}\frac{1}{d(u,u_{i})+t}+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in \Theta_{i}}\frac{1}{d(v,u_{i})+t},\)
7. \(RCW(G)=\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}\frac{1}{D(G)+1-d(u,u_{i})}+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in \Theta_{i}}\\ \frac{1}{D(G)+1-d(v,u_{i})},\)
8. \(\pi(G)=\prod_{i=1}^{r}\prod_{j=i+1}^{r}\prod_{u\in\Theta_{j}}d^{|\Theta_{i}|}(u,u_{i})\times\sqrt{\prod_{i=1}^{r}\prod_{v\in \Theta_{i}}d^{|\Theta_{i}|}(v,u_{i})},\)
9. \(\Pi(G)=\ln\sqrt{2\prod_{i=1}^{r}\prod_{j=i+1}^{r}\prod_{u\in\Theta_{j}}d^{|\Theta_{i}|}(u,u_{i})\times\sqrt{\prod_{i=1}^{r}\prod_{v\in \Theta_{i}}d^{|\Theta_{i}|}(v,u_{i})}}.\)

Proof. We only prove for \(W_{\lambda}(G)\). The remaining cases can be proved in similar fashion. Since \(W_{\lambda}(u,G)\) is equal to the sum of all vertices in the same orbit, we infer \begin{eqnarray*} \sum_{w\in V(G)}W_{\lambda}(w,G)&=&\sum_{i=1}^{r}\sum_{w\in \Theta_{i}}W_{\lambda}(w,G)\\&=&\sum_{i=1}^{r}|\Theta_{i}|W_{\lambda}(u_{i},G)\\ &=&\sum_{i=1}^{r}|\Theta_{i}|\sum_{j=1}^{r}\sum_{y\in \Theta_{i}}d^{\lambda}(u_{i},y)\\&=&\sum_{i=1}^{r}\sum_{j=1}^{r}|\Theta_{i}|\sum_{y\in \Theta_{i}}d^{\lambda}(u_{i},y)\\ &=&2\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{y\in\Theta_{i}}d^{\lambda}(y,u_{i})+\sum_{i=1}^{r}|\Theta_{i}|\sum_{z\in \Theta_{i}}d^{\lambda}(z,u_{i}). \end{eqnarray*} Hence, in terms of (1), \begin{eqnarray*}W_{\lambda}(G)&=&\frac{1}{2}\sum_{w\in V(G)}W_{\lambda}(w,G)\\ &=&\sum_{i=1}^{r}\sum_{j=i+1}^{r}|\Theta_{i}|\sum_{u\in\Theta_{j}}d^{\lambda}(u,u_{i})+\frac{1}{2}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in \Theta_{i}}d^{\lambda}(v,u_{i}). \end{eqnarray*} Hence, the desired result is obtained.

The next result is about the computation of topological indices when the point group of the molecular graph is transitive.

Lemma 2.2. If the point group \(P_{G}\) of the molecular graph is transitive. Then for any \(v\in V(G)\), we have

1. \(W_{\lambda}(G)=\frac{|V(G)|}{2}W_{\lambda}(v,G),\)
2. \(WW(G)=\frac{|V(G)|}{4}WW(v,G),\)
3. \(WW_{\lambda}(G)=\frac{|V(G)|}{4}WW_{\lambda}(v,G),\)
4. \(H(G)=\frac{|V(G)|}{2}H(v,G),\)
5. \(H(G,x)=\frac{|V(G)|}{2}H(v,G,x),\)
6. \(H_{t}(G)=\frac{|V(G)|}{2}H_{t}(v,G),\)
7. \(RCW(G)=\frac{|V(G)|}{2}RCW(v,G),\)
8. \(\pi(G)=\sqrt{\pi^{|V(G)|}(v,G)},\)
9. \(\Pi(G)=\ln\sqrt{2\sqrt{\pi^{|V(G)|}(v,G)}}.\)

In terms of Lemma 2.2, to calculate the distance-based topological indices of the molecular graph with transitive point group, we only need to choose any vertex \(v\in V(G)\) and calculate the distances between \(v\) and \(u\in V(G)-\{v\}\). Take a subgroup \(H\) of \(P_{G}\), which is not necessarily transitive even if \(P_{G}\) is transitive. Now, the vertex set \(V(G)\) can be divided into orbits of \(H\) such that \(\Theta_{1},\Theta_{2},\cdots,\Theta_{r}\) with \((|\Theta_{1}|\le|\Theta_{2}|\le\cdots|\Theta_{r}|\).

Theorem 2.3. Let \(v_{i}\in \Theta_{i}\), \(i=1,2,\cdots,r\). Then,

1. \(W_{\lambda}(v_{1},G)=\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}|\Theta_{i}|d^{\lambda}(u,v_{i}),\)
2. \(WW(v_{1},G)=\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\frac{|\Theta_{i}|}{2}(d(u,v_{i})+d^{2}(u,v_{i})),\)
3. \(WW_{\lambda}(v_{1},G)=\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\frac{|\Theta_{i}|}{2}(d^{\lambda}(u,v_{i})+d^{2\lambda}(u,v_{i})),\)
4. \(H(v_{1},G)=\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\frac{|\Theta_{i}|}{d(u,v_{i})},\)
5. \(H(v_{1},G,x)=\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\frac{|\Theta_{i}|}{d(u,v_{i})}x^{d(u,v_{i})},\)
6. \(H_{t}(v_{1},G)=\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\frac{|\Theta_{i}|}{d(u,v_{i})+t},\)
7. \(RCW(v_{1},G)=\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\frac{|\Theta_{i}|}{D(G)+1-d(u,v_{i})}.\)

Proof. Since \(P_{G}\) is transitive, we get \begin{eqnarray*} W_{\lambda}(v_{1},G)&=&\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{z\in V(G)}d^{\lambda}(u,z)\\ &=&\frac{1}{|\Theta_{1}|}\sum_{u\in \Theta_{1}}\sum_{i=1}^{r}\sum_{z\in \Theta_{i}}d^{\lambda}(u,z)\\ &=&\frac{1}{|\Theta_{1}|}\sum_{i=1}^{r}\sum_{z\in \Theta_{i}}\sum_{u\in \Theta_{1}}d^{\lambda}(u,z). \end{eqnarray*} For any \(i\) and any \(z\in \Theta_{i}\), there exist \(h_{i}\in H\) satisfying \(h_{i}(z)=v_{i}\). Therefore, \begin{eqnarray*} \sum_{z\in \Theta_{i}}\sum_{u\in \Theta_{1}}d^{\lambda}(u,z)&=&\sum_{z\in \Theta_{i}}\sum_{u\in \Theta_{1}}d^{\lambda}(h_{i}(u),h_{i}(z))\\ &=&\sum_{z\in \Theta_{i}}\sum_{h_{i}(u)\in \Theta_{1}}d^{\lambda}(h_{i}(u),v_{i})\\&=&|\Theta_{i}|\sum_{h_{i}(u)\in \Theta_{1}}d^{\lambda}(h_{i}(u),v_{i})\\ &=&|\Theta_{i}|\sum_{v\in\Theta_{1}}d^{\lambda}(v,v_{i}). \end{eqnarray*} Consequently, we yield \begin{eqnarray*} W_{\lambda}(v_{1},G)&=&\frac{1}{|\Theta_{1}|}\sum_{i=1}^{r}\sum_{u\in \Theta_{1}}\sum_{z\in \Theta_{i}}d^{\lambda}(u,z)\\ &=&\frac{1}{|\Theta_{1}|}\sum_{i=1}^{r}|\Theta_{i}|\sum_{v\in\Theta_{1}}d^{\lambda}(v,v_{i})\\ &=&\frac{1}{|\Theta_{1}|}\sum_{v\in\Theta_{1}}\sum_{i=1}^{r}|\Theta_{i}|d^{\lambda}(v,v_{i}). \end{eqnarray*} The remaining parts follows similarly, hence, we complete the proof.

In theorem 2.3, we can see that for a vertex \(v_{1}\in\Theta_{1}\), we do not need to compute all distances between \(v_{1}\) and \(V(G)-\{v_{1}\}\). It is enough to select one vertex from \(\Theta_{i}\). In real practice, we select a subgroup \(H\) so that \(\Theta_{1}\) is as small as possible in order to simplify the calculation. Specially, if \(|\Theta_{1}|=1\) (\(H\) fixes \(v_{1}\)), we only count \(r-1\) times. Hence, we can give the following corollary.

Corollary 2.4. Let \(v_{i}\in\Theta_{i}\), \(i=1,2,\cdots,r\). Assume that \(|\Theta_{1}|=1\). We get

1. \(W_{\lambda}(v_{1},G)=\sum_{i=2}^{r}|\Theta_{i}|d^{\lambda}(v_{1},v_{i}),\)
2. \(WW(v_{1},G)=\sum_{i=2}^{r}|\Theta_{i}|\frac{d(v_{1},v_{i})+d^{2}(v_{1},v_{i})}{2},\)
3. \(WW_{\lambda}(v_{1},G)=\sum_{i=2}^{r}|\Theta_{i}|\frac{d^{\lambda}(v_{1},v_{i})+d^{2\lambda}(v_{1},v_{i})}{2},\)
4. \(H(v_{1},G)=\sum_{i=2}^{r}|\Theta_{i}|\frac{1}{d(v_{1},v_{i})},\)
5. \(H(v_{1},G,x)=\sum_{i=2}^{r}|\Theta_{i}|\frac{1}{d(v_{1},v_{i})}x^{d(v_{1},v_{i})},\)
6. \(H_{t}(v_{1},G)=\sum_{i=2}^{r}|\Theta_{i}|\frac{1}{d(v_{1},v_{i})+t},\)
7. \(RCW(v_{1},G)=\sum_{i=2}^{r}|\Theta_{i}|\frac{1}{D(G)+1-d(v_{1},v_{i})},\)
8. \(\pi(v_{1},G)=\prod_{i=2}^{r}d^{|\Theta_{i}|}(v_{1},v_{i}),\)
9. \(\Pi(v_{1},G)=\ln\sqrt{2\prod_{i=2}^{r}d^{|\Theta_{i}|}(v_{1},v_{i})}.\)

In order to reduce the computation steps of the distance-based topological indices, note that a large number of the molecular structures have the layered structure such that the different orbits have consecutive distances from a fixed vertex. In such a situation, we have following theorem.

Theorem 2.5. Assume that \(P_{G}\) is transitive and \(H\unlhd P_{G}\) is a subgroup with orbits \(\Theta_{1},\Theta_{2},\cdots,\Theta_{r}\) such that \(|\Theta_{i}|=1\). Let \(v_{i}\in\Theta_{i}\) for \(i=1,2,\cdots,r\). Then we have

1. \(W_{\lambda}(G)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|(j-1)^{\lambda},\)
2. \(WW(G)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{j^{2}-j}{2},\)
3. \(WW_{\lambda}(G)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{(j-1)^{\lambda}+(j-1)^{2\lambda}}{2},\)
4. \(H(G)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1},\)
5. \(H(G,x)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}x^{j-1},\)
6. \(H_{t}(G)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j+t-1},\)
7. \(RCW(G)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{D(G)+2-j},\)
8. \(\pi(G)=(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}},\)
9. \(\Pi(G)=\ln\sqrt{(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}}.\)

Proof. Science the molecular graphs are connected and all the elements in the same orbit have equal distances from \(v_{1}\), the orbits saturate the vacancy between \(\Theta_{1}\) and \(\Theta_{k}\) by means of their distances from \(v_{1}\). Since only \(r-1\) orbits different from \(\Theta_{1}\) and \(d(v_{1},v_{k})\ge r-1\), we infer that the orbits run consecutively between \(\Theta_{1}\) and \(\Theta_{k}\), which reveals that the vertices \(v_{1},v_{2},\cdots,v_{r}\) can be permuted into \(v_{k_{1}},v_{k_{2}},\cdots,v_{k_{r}}\) with \(k_{1}=1\) and \(k_{r}=k\) satisfies \(d(v_{1},v_{k_{j}})=j-1\), \(j=1,2,\cdots,r\). Therefore, \begin{eqnarray*} W_{\lambda}(G)&=&\frac{n}{2}W_{\lambda}(v_{1},G)\\&=&\frac{n}{2}\sum_{i=2}^{r}|\Theta_{i}|d^{\lambda}(v_{1},v_{i})\\ &=&\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|d^{\lambda}(v_{1},v_{k_{j}})\\&=&\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|(j-1)^{\lambda}. \end{eqnarray*} The remaining cases can be easily proved in similar fashion.

For the presentation of examples in the next section, we should mention that all the regular polyhedrons meet the conditions in the Theorem 2.5.

3. Computation Examples

In this section, we give five illustrative examples to explain our method. In the following contexts, we always assume that $n$ is the number of vertex in molecular graph $G$ and the regular polyhedrons meet the conditions of the theorem 2.5.

Example 3.1. [Computation on tetrahedron] The structure of tetrahedron (denoted by \(G_{1}\)) can refer to Figure 1. Let \(P_{G_{1}}\) be its point group. We need first determine the subgroup \(R\unlhd P_{G_{1}}\) of all the rotation in \(P_{G_{1}}\). The elements consisting of \(R\) are as follows: (1) the identity; (2) rotations through the angle \(\pi\) about each of three axes joining the midpoints of opposite edges; (3) rotations through angles of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) on the each of four axes joining vertices with centers of opposite faces. So, we have \(|R|=12\).

Clearly, \(R\) and \(P_{G_{1}}\) are transitive. Select \(H\) as the identity plus the set of all rotations around the axis joining \(v_{1}\) with the center of the opposite face through angles of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) anticlockwise. We yield two orbits with representatives \(v_{1}\), \(v_{2}\) as presented in the Figure 1. Applying \(|\Theta_{i}|=\frac{|P|}{|P_{v}|}\), we infer \(|\Theta_{1}|=1\), \(|\Theta_{2}|=3\). According to theorem 2.5, we get

1. \(W_{\lambda}(G_{1})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|(j-1)^{\lambda}=\frac{4}{2}\cdot3=6,\)
2. \(WW(G_{1})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{j^{2}-j}{2}=\frac{4}{2}\cdot3\cdot\frac{1}{2}(1+1^{2})=6,\)
3. \(WW_{\lambda}(G_{1})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{(j-1)^{\lambda}+(j-1)^{2\lambda}}{2}=\frac{4}{2}\cdot3\cdot\frac{1}{2}(1^{\lambda}+1^{2\lambda})=6,\)
4. \(H(G_{1})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}=\frac{4}{2}\cdot3=6,\)
5. \(H(G_{1},x)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}x^{j-1}=\frac{4}{2}\cdot3\cdot x=6x,\)
6. \(H_{t}(G_{1})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j+t-1}=\frac{4}{2}\cdot3\cdot\frac{1}{1+t}=\frac{6}{1+t},\)
7. \(RCW(G_{1})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{D(G)+2-j}=\frac{4}{2}\cdot3\cdot\frac{1}{1+1-1}=6,\)
8. \(\pi(G_{1})=(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}=1,\)
9. \(\Pi(G_{1})=\ln\sqrt{(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}}=\ln\sqrt{2}.\)

Example 3.2. [Computation on cube] \noindent The structure of cube (denoted as \(G_{2}\)) can refer to Figure 2. In this case, the subgroup \(R\unlhd P_{G_{2}}\) of all the rotations consists of the follows: (1) rotations through the angle \(\pi\) on each of six axes joining midpoints of diagonally opposite edges; (2) rotations through angles of \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) about each of four axes joining extreme opposite vertices; (3) rotations through angles of \(\frac{\pi}{2},\pi\), and \(\frac{3\pi}{2}\) about each of three axes joining the centers of opposite faces. Thus, by simple computation, we get \(|R|=24\).

Clearly, \(R\) and \(P_{G_{2}}\) are both transitive. \(H\) is selected as in the first instance but the rotations are around the axis joining the two opposite vertices \(v_{1}\) and \(v_{3}\). We get four orbits with representatives \(v_{1}\), \(v_{2}\), \(v_{3}\) and \(v_{4}\) as presented in the Figure 2. In view of \(|\Theta_{i}|=\frac{|P|}{|P_{v}|}\), we yield \(|\Theta_{1}|=|\Theta_{4}|=1\), \(|\Theta_{2}|=|\Theta_{3}|=3\). Applying theorem 2.5, we get

1. \(W_{\lambda}(G_{2})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|(j-1)^{\lambda}=\frac{8}{2}(3+3\cdot2^{\lambda}+3^{\lambda})=12+12\cdot2^{\lambda}+4\cdot3^{\lambda},\)
2. \(WW(G_{2})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{j^{2}-j}{2}=\frac{8}{2}(\frac{3}{2}(1+1)+\frac{3}{2}(2+4)+\frac{3+9}{2})=72,\)
3. \(WW_{\lambda}(G_{2})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{(j-1)^{\lambda}+(j-1)^{2\lambda}}{2}=\frac{8}{2}(\frac{3}{2}(1+1) +\frac{3}{2}(2^{\lambda}+4^{\lambda})+\frac{3^{\lambda}+9^{\lambda}}{2}) =12+6(2^{\lambda}+4^{\lambda})+2(3^{\lambda}+9^{\lambda}),\)
4. \(H(G_{2})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}=\frac{8}{2}(3+\frac{3}{2}+\frac{1}{3})=\frac{58}{3},\)
5. \(H(G_{2},x)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}x^{j-1}=\frac{8}{2}(3x+\frac{3}{2}x^{2}+\frac{1}{3}x^{3})=12x+6x^{2}+\frac{4}{3}x^{3},\)
6. \(H_{t}(G_{2})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j+t-1}=4(\frac{3}{1+t}+\frac{3}{2+t}+\frac{1}{3+t}),\)
7. \(RCW(G_{2})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{D(G)+2-j}=\frac{8}{2}(\frac{3}{3}+\frac{3}{2}+\frac{1}{1})=14,\)
8. \(\pi(G_{2})=(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}=(1^{3}\cdot2^{3}\cdot3)^{\frac{8}{2}}=331776,\)
9. \(\Pi(G_{2})=\ln\sqrt{(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}}=\ln576\sqrt{2}.\)

Example 3.3. [Computation on octahedron] The structure of octahedron (denoted by \(G_{3}\)) can refer to Figure 3. Obviously, we can get the octahedron by adding the midpoints of adjacent faces of the cube with edges. Form this point of view, its point group is the same as that of the cube.

Furthermore, \(P_{G_{3}}\) is transitive, and all the rotations can be selected around the axis joining \(v_{1}\) and \(v_{3}\) which keep the octahedron invariant. Three orbits are obtained with representatives \(v_{i}\), \(i=1,2,3\) as depicted in the Figure 3 and \(|\Theta_{1}|=|\Theta_{3}|=1\) and \(|\Theta_{2}|=4\). By theorem 2.5, we get

1. \(W_{\lambda}(G_{3})=12+3\cdot2^{\lambda},\)
2. \(WW(G_{3})=21,\)
3. \(WW_{\lambda}(G_{3})=12+\frac{3}{2}(2^{\lambda}+4^{\lambda}),\)
4. \(H(G_{3})=\frac{27}{2},\)
5. \(H(G_{3},x)=12x+3x^{2},\)
6. \(H_{t}(G_{3})=\frac{12}{1+t}+\frac{3}{2+t},\)
7. \(RCW(G_{3})=9,\)
8. \(\pi(G_{3})=8,\)
9. \(\Pi(G_{3})=\ln4.\)

Example 3.4. [Computation on icosahedron] The structure of icosahedron (denoted by \(G_{4}\)) can refer to Figure 4. The rotation subgroup \(R\) of the point group consists: (1)the identity; (2) rotations through the angle \(\pi\) about each of fifteen axes joining midpoints of opposite edges; (3) rotations through angles of \(\frac{2\pi}{3}\) and \(\frac{4\pi}{3}\) about each of ten axes joining centers of opposite faces; (4) rotations through angles of \(\frac{2\pi}{5},\frac{4\pi}{5},\frac{6\pi}{5}\), and \(\frac{8\pi}{5}\) about each of six axes joining extreme opposite vertices. Therefore, we have \(|R|=60\).

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Furthermore, \(R\) and \(P_{G_{4}}\) are transitive. \(H\) can be selected as the group generated by the \(\frac{2\pi}{5}\)-rotation around the axis joining \(v_{1}\) and \(v_{4}\). There are four orbits with representatives as shown in the Figure 4 and by \(|\Theta_{i}|=\frac{|P|}{|P_{v}|}\) we get \(|\Theta_{1}|=|\Theta_{4}|=1\), \(|\Theta_{2}|=|\Theta_{3}|=5\). Applying theorem 2.5, we get

1. \(W_{\lambda}(G_{4})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|(j-1)^{\lambda}=\frac{12}{2}(5+5\cdot2^{\lambda}+3^{\lambda})=30+30\cdot2^{\lambda}+6\cdot3^{\lambda},\)
2. \(WW(G_{4})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{j^{2}-j}{2}=\frac{12}{2}(\frac{5}{2}(1+1)+\frac{5}{2}(2+4)+\frac{3+9}{2})=156,\)
3. \(WW_{\lambda}(G_{4})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{(j-1)^{\lambda}+(j-1)^{2\lambda}}{2}=\frac{12}{2} (\frac{5}{2}(1+1)+\frac{5}{2}(2^{\lambda}+4^{\lambda})+\frac{3^{\lambda}+9^{\lambda}}{2}) =30+15(2^{\lambda}+4^{\lambda})+3(3^{\lambda}+9^{\lambda}),\)
4. \(H(G_{4})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}=\frac{12}{2}(5+\frac{5}{2}+\frac{1}{3})=47,\)
5. \(H(G_{4},x)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}x^{j-1}=\frac{12}{2}(5x+\frac{5}{2}x^{2}+\frac{1}{3}x^{3})=30x+15x^{2}+2x^{3},\)
6. \(H_{t}(G_{4})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j+t-1}=6(\frac{5}{1+t}+\frac{5}{2+t}+\frac{1}{3+t}),\)
7. \(RCW(G_{4})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{D(G)+2-j}=\frac{12}{2}(\frac{5}{3}+\frac{5}{2}+\frac{1}{1})=31,\)
8. \(\pi(G_{4})=(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}=(1^{5}\cdot2^{5}\cdot3)^{\frac{12}{2}}=96^{6},\)
9. \(\Pi(G_{4})=\ln\sqrt{(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}}=\ln96^{3}\sqrt{2}.\)

Example 3.5. [Computation on dodecahedron] The structure of dodecahedron (denoted by \(G_{5}\)) can refer to Figure 5. Similar as we discussed in the above examples, one can see that the dodecahedron and icosahedron have the same point group. Hence, \(H\) can be selected to be the group generated by the \(\frac{2\pi}{3}\)-rotation around the axis joining \(v_{1}\) and \(v_{6}\), and the reflection with respect to the plane containing \(v_{1},v_{2}\) and \(v_{6}\).

There are six orbits with representatives \(v_{i}\), \(i=1,2,\cdots,6\) as shown in the Figure 5. By simple computation, we get \(|\Theta_{1}|=|\Theta_{6}|=1\), \(|\Theta_{2}|=|\Theta_{5}|=3\) and \(|\Theta_{3}|=|\Theta_{4}|=6\). Thus, in terms of Theorem 2.5, we get

1. \( W_{\lambda}(G_{5})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|(j-1)^{\lambda}=\frac{20}{2}(3+6\cdot2^{\lambda}+6\cdot3^{\lambda}+3\cdot4^{\lambda}+5^{\lambda}) =30+60\cdot2^{\lambda}+60\cdot3^{\lambda}+30\cdot4^{\lambda}+10\cdot5^{\lambda},\)
2. \( WW(G_{5})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{j^{2}-j}{2} =\frac{20}{2}(\frac{3}{2}(1+1)+3(2+4)+3(3+9)+\frac{3}{2}(4+16)+\frac{1}{2}(5+25))=1020,\)
3. \( WW_{\lambda}(G_{5})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{(j-1)^{\lambda}+(j-1)^{2\lambda}}{2} =\frac{20}{2}(\frac{3}{2}(1+1)+3(2^{\lambda}+4^{\lambda})+3(3^{\lambda}+9^{\lambda})+\frac{3}{2}(4^{\lambda}+16^{\lambda})+\frac{1}{2}(5^{\lambda}+25^{\lambda})) =30+30(2^{\lambda}+4^{\lambda})+30(3^{\lambda}+9^{\lambda})+15(4^{\lambda}+16^{\lambda})+5(5^{\lambda}+25^{\lambda}),\)
4. \( H(G_{5})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1} =\frac{20}{2}(3+\frac{6}{2}+\frac{6}{3}+\frac{3}{4}+\frac{1}{5})=\frac{179}{2},\)
5. \( H(G_{5},x)=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j-1}x^{j-1}=\frac{20}{2}(3x+\frac{6}{2}x^{2}+\frac{6}{3}x^{3}+\frac{3}{4}x^{4}+\frac{1}{5}x^{5}) =30x+30x^{2}+20x^{3}+\frac{15}{2}x^{4}+2x^{5},\)
6. \( H_{t}(G_{5})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{j+t-1}=\frac{20}{2}(\frac{3}{1+t}+\frac{6}{2+t}+\frac{6}{3+t}+\frac{3}{4+t}+\frac{1}{5+t}) =\frac{30}{1+t}+\frac{60}{2+t}+\frac{60}{3+t}+\frac{30}{4+t}+\frac{10}{5+t},\)
7. \(RCW(G_{5})=\frac{n}{2}\sum_{j=2}^{r}|\Theta_{k_{j}}|\frac{1}{D(G)+2-j} =\frac{20}{2}(\frac{3}{5}+\frac{6}{4}+\frac{6}{3}+\frac{3}{2}+\frac{1}{1})=66,\)
8. \(\pi(G_{5})=(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}=(2^{6}3^{6}4^{3}5)^{10},\)
9. \(\Pi(G_{5})=\ln\sqrt{(\prod_{j=2}^{r}(j-1)^{|\Theta_{k_{j}}|})^{\frac{n}{2}}}=\ln\sqrt{2^{61}3^{60}4^{30}5^{10}}.\)

4. Conclusion

In this paper, we mainly report the approach on how to use group theory to determine the distance-based topological indices for certain important symmetry chemical structures. Since these Wiener related and other distance-based topological indices are widely applied in the analysis of both the boiling point and melting point of chemical compounds and QSPR/QSAR study, the promising prospects of their application for the chemical, medical and pharmacy engineering will be illustrated in the theoretical conclusion that is obtained in this article.

Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

When delays appear in additional terms involving the highest order derivative of the unknown function in a differential equation, we are dealing with a neutral type differential equation. Neutral functional differential equations have numerous applications in electric networks. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines which rise in high speed computers where the lossless transmission lines are used to interconnect switching circuits; see [1]. Recently, many results on oscillation of nonneutral differential equations and neutral functional differential equations have been established. We refer the reader to [2, 4, 4] and the references cited therein.

The theory of time scales is initiated by Hilger [5, 6] in order to unify continuous and discrete analysis. Several authors have expounded on various aspects of the theory of dynamic equations on time scales; see the survey paper by Agarwal et al. [7] and the references cited therein. The books on the subject of time scales, by Bohner and Peterson [8], summarize and organize much of time scale calculus. There are applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, etc.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of ordinary dynamic equations on time scales, we refer the reader to the papers [9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Recently Agarwal et al. [24] have established some new oscillation criteria for second-order delay dynamic equations on time scales.

Very recently, some authors studied on existence and behavior of solutions for some integral equations, second order multi objective symmetric programming problem and duality relations and fixed point theorems for nonlinear contractions in [25, 26, 27, 28, 29].

In this paper, we consider second-order nonlinear neutral dynamic equations of the following form:

On a time scale \(\mathbb{T}\) satisfying \(\inf\mathbb{T}=t_{0}\) and \(\sup\mathbb{T}=\infty\). We recall that a solution of equation (1) is said to be oscillatory on \([t_{0},\infty)_{\mathbb{T}}\) in case it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. Equation (1) is said to be oscillatory in case all of its solutions are oscillatory.

Throughout this paper, we assume the followings:
\((H_{1})\hspace{.2cm}\tau(t), \delta_{i}(t)\in C_{rd}(\mathbb{T},\mathbb{T})\) such that \(\tau(t)\leqslant t \) and \(\displaystyle\lim_{t\rightarrow\infty}\tau(t)= \displaystyle\lim_{t\rightarrow\infty}\delta_{i}(t)=\infty,\hspace{.2cm}\hspace{.2cm}i=1,2\),
\((H_{2})\hspace{.2cm}r(t)\in C_{rd}(\mathbb{T},\mathbb{R}^{+})\) such that \(\int_{t_{0}}^{\infty}\frac{1}{r^{\frac{1}{\gamma}}(t)}\Delta t=\infty \) and \(m(t)\in C_{rd}(\mathbb{T},\mathbb{R}^{+})\), \(p(t)\in C_{rd}(\mathbb{T},[0,1))\) such that \(m(\tau(t))>p(t)\) where \(\mathbb{R}^{+}=[0,\infty)\),
\((H_{3})\hspace{.2cm}f_{i}(t,u):\mathbb{T}\times\mathbb{R}\longrightarrow\mathbb{R}\hspace{.2cm}\) are continuous functions such that \(uf_{i}(t,u)>0\) for all \(u\neq0\) and there exist \(q_{i}(t)\in C_{rd}(\mathbb{T},\hspace{.1cm}\mathbb{R}^{+})\hspace{.1cm}(i=1,2), \gamma\) is quotients of odd positive integers \(\alpha\) and \(\beta \) with \(0< \alpha\leq\beta\) such that \(|uf_{1}(t,u)|\geq q_{1}(t)|u|^{\alpha+1}, \hspace{.1cm}|uf_{2}(t,u)|\geq q_{2}(t)|u|^{\beta+1}\).
This paper is organized as follows. After this introduction, we introduce some basic lemmas in Section 2. In Section 3, we present the main results and give an example to illustrate the main results.

2. Some preliminaries

Before stating our main results, we'll give some lemmas which play an important role in the proof of the main results. Set then the equation (1) becomes For \(t,T\in\mathbb{T}\) with \(t>T\), we define \begin{equation*} R(t,T)=\int_{T}^{t}\frac{1}{(r(s))^\frac{1}{\gamma}}\Delta s, \end{equation*} \begin{eqnarray*} \beta(t,T)=\left\{ \begin{array}{ll} \frac{R(\delta_{i}(t),T)}{R(t,T)}, & \hbox{$\delta_{i}(t)< t$;} \\ \ \ 1 , & \hbox{$\delta_{i}(t)\geq t $,} \end{array} \right. \end{eqnarray*} \begin{eqnarray*} \eta^{\sigma}(t)=\left\{ \begin{array}{ll} 1, & \hbox{$\alpha=\gamma$;} \\ c_{2}^{\alpha-\gamma}\bigg(\int_{T}^{\sigma(t)} \frac{1}{r^{\frac{1}{\gamma}}(s)}\Delta s\bigg)^{\alpha-\gamma}, & \hbox{$\alpha< \gamma$;} \\ c_{1}, & \hbox{$\alpha>\gamma $,} \end{array} \right. \end{eqnarray*} and \begin{equation*} Q_{1}(t)= Q(t)\bigg(\frac{r^{\frac{1}{\gamma}}(t)R(t,T)}{r^{\frac{1}{\gamma}}(t)R(t,T)+\mu(t)}\bigg)^{\alpha}\eta^{\sigma}(t), \end{equation*} where \(Q(t)\) will be defined as Lemma 2.2. For \(D=\{{(t,s)\in \mathbb{T}^{2}:t\geq s\geq 0\}}\), we define \( \mathcal{H}=\{H(t,s)\in C_{rd}^{1}(D,[0,\infty)):H(t,t)=0,H(t,s)>0 \ \ \textrm{and} \ \ H_{s}^{\Delta}(t,s)\geq 0 \ \ \textrm{for} \ \ t>s\geq 0\} \) and \begin{equation*} C(t,s)=H^{\Delta}(t,s)z(s)+H(t,\sigma(s))z^{\Delta}(s) \ \ \textrm{for} \ \ H(t,s)\in\mathcal{H}, \end{equation*} where \(z\in C_{rd}^{1}(\mathbb{T},(0,\infty))\) is to be given Theorem 3.2 and Theorem 3.3, and \(z_{+}^{\Delta}(t)=max\{z^{\Delta}(t),0\}\).

Lemma 2.1. Assume that \((H_{1})-(H_{3})\) and (1) are satisfied. If equation (1) has a nonoscillatory solutions \(y\) on \([t_{0},\infty)_{\mathbb{T}}\), and \(x\) is defined as in (2), then there exists a \(T\in \mathbb{T}\) sufficiently large such that \(x(t)>0,\hspace{.1cm} x^{\Delta}(t)>0,\) \((r(t)(x^{\Delta}(t))^{\gamma})^{\Delta}< 0, \hspace{.1cm} x(t)\geq r^{\frac{1}{\gamma}}(t)x^{\Delta}(t)R(t,T),\hspace{.1cm} x(\delta_{i}(t))\geq \beta(t,T)x(t)\hspace{.1cm} for\hspace{.1cm} t\in [T,\infty)_{\mathbb{T}}.\)

Proof. If \(y(t)\) is an eventually positive solution of (1), then there exist a \(T\in [t_{0},\infty)_{\mathbb{T}}\) such that

From (2), (4) and \((H_{2}),\hspace{.1cm}x(t)>0\). Also by (3) and \((H_{3})\), we have \begin{eqnarray*} (r(t)(x^{\Delta}(t))^{\gamma})^{\Delta}&=& \nonumber -f_{1}(t,y(\delta_{1}(t))-f_{2}(t,y(\delta_{2}(t))\\ &\leq&-q_{1}(t)y^{\alpha}(\delta_{1}(t))-q_{2}(t)y^{\beta}(\delta_{2}(t))<0,\hspace{.2cm}\textrm{for} \hspace{.2cm}t\geq T, \end{eqnarray*} which implies that \(r(t)(x^{\Delta}(t))^{\gamma}\) is decreasing on \([T,\infty)_{\mathbb{T}}\).
We claim that \(r(t)(x^{\Delta}(t))^{\gamma}>0 \; \; \textrm{on}\; \; [T,\infty)_{\mathbb{T}}\). Assume not, there is a \(t_{1}\in[T,\infty)_{\mathbb{T}}\) such that \( r(t_{1})(x^{\Delta}(t_{1}))^{\gamma}=c< 0\). Since \(r(t)(x^{\Delta}(t))^{\gamma}\) is decreasing on \([T,\infty)_{\mathbb{T}}\), \(r(t)(x^{\Delta}(t))^{\gamma}\leq r(t_{1})(x^{\Delta}(t_{1}))^{\gamma}=c \) for \(t\geq t_{1}\). So, we have \begin{eqnarray*} x^{\Delta}(t) \leq \frac{c^\frac{1}{\gamma}}{r^{\frac{1}{\gamma}}(t)}. \end{eqnarray*} Integrating the above inequality from \(t_{1}\) to t, by \((H_{2})\), we get \begin{eqnarray*} x(t) \leq x(t_{1})+c^{\frac{1}{\gamma}}\int_{t_{1}}^{t}\frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s\rightarrow -\infty \ \ \ (t\rightarrow\infty) \end{eqnarray*} and this contradicts the fact that \(x(t)>0\) for all \(t\geq T\). Thus, we have \(r(t)(x^{\Delta}(t))^{\gamma}>0\) on \([T,\infty)_{\mathbb{T}}\) and so \(x^{\Delta}(t)>0\) on \([T,\infty)_{\mathbb{T}}\). From \(r(t)(x^{\Delta}(t))^{\gamma}\) is decreasing on \([T,\infty)_{\mathbb{T}}\), we have \begin{eqnarray*} x(t)&=& \nonumber x(T)+\int_{T}^{t}x^{\Delta}(s)\Delta s\\ &=& \nonumber x(T)+\int_{T}^{t} \frac{(r(s)(x^{\Delta}(s))^{\gamma})^{\frac{1}{\gamma}}}{r^{\frac{1}{\gamma}}(s)} \Delta s\\ &>& \nonumber \int_{T}^{t} \frac{(r(s)(x^{\Delta}(s))^{\gamma})^{\frac{1}{\gamma}}}{r^{\frac{1}{\gamma}}(s)} \Delta s\\ &>&\nonumber (r(t)(x^{\Delta}(t))^{\gamma})^{\frac{1}{\gamma}}\int_{T}^{t}\frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s\\ &=&r^{\frac{1}{\gamma}}(t)x^{\Delta}(t)R(t,T). \end{eqnarray*} Now, we will show that \begin{eqnarray*} x(\delta_{i}(t))\geq \beta(t,T)x(t). \end{eqnarray*} We consider two cases which \(\delta_{i}(t)< t\) and \(\delta_{i}(t)\geq t\), respectively.
\(\mathbf{Case \hspace{.1cm}1.}\quad \delta_{i}(t)< t\). Since \(r(t)(x^{\Delta}(t))^{\gamma}\) is decreasing on \([T,\infty)_{\mathbb{T}}\), we have \begin{eqnarray*} x(t)-x(\delta_{i}(t))&=&\int_{\delta_{i}(t)}^{t} \frac{(r(s)(x^{\Delta}(s))^{\gamma})^{\frac{1}{\gamma}}}{r^{\frac{1}{\gamma}}(s)} \Delta s \\&\leq&(r(\delta_{i}(t))(x^{\Delta}(\delta_{i}(t)))^{\gamma})^{\frac{1}{\gamma}} \int_{\delta_{i}(t)}^{t} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s. \end{eqnarray*} By dividing \(x(\delta_{i}(t))\), it follows that \begin{eqnarray*} \frac{x(t)}{x(\delta_{i}(t))}\leq 1+ \frac{(r(\delta_{i}(t))(x^{\Delta}(\delta_{i}(t)))^{\gamma})^{\frac{1}{\gamma}}}{x(\delta_{i}(t))} \int_{\delta_{i}(t)}^{t} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s. \end{eqnarray*} Since \(\delta_{i}(t)\geq T\) for \(t\in[T,\infty)_{\mathbb{T}}\), we have \begin{eqnarray*} x(\delta_{i}(t))\geq \int_{T}^{\delta_{i}(t)} \frac{(r(s)(x^{\Delta}(s))^{\gamma})^{\frac{1}{\gamma}}}{r^{\frac{1}{\gamma}}(s)} \Delta s \geq(r(\delta_{i}(t))(x^{\Delta}(\delta_{i}(t)))^{\gamma})^{\frac{1}{\gamma}}\int_{T}^{\delta_{i}(t)} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s, \end{eqnarray*} which implies that \begin{eqnarray*} \frac{(r(\delta_{i}(t))(x^{\Delta}(\delta_{i}(t)))^{\gamma})^{\frac{1}{\gamma}}}{x(\delta_{i}(t))}\leq \frac{1}{\int_{T}^{\delta_{i}(t)} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s}. \end{eqnarray*} Thus, we get \begin{eqnarray*} \frac{x(t)}{x(\delta_{i}(t))}\leq 1+\frac{\int_{\delta_{i}(t)}^{t} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s} {\int_{T}^{\delta_{i}(t)} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s} =\frac{\int_{T}^{t} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s}{\int_{T}^{\delta_{i}(t)} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s} =\frac{R(t,T)}{R(\delta_{i}(t),T)}. \end{eqnarray*} So, \begin{eqnarray*} x(\delta_{i}(t))\geq\beta(t,T)x(t). \end{eqnarray*} \(\mathbf{Case\hspace{.1cm} 2.}\quad \delta_{i}(t)\geq t\). Since \(x^{\Delta}(t)\geq0\) for \(t\in[T,\infty)_{\mathbb{T}}, \ \ x(\delta_{i}(t))\geq x(t)\). From the definition of \(\beta(t,T)\), we obtain \begin{eqnarray*} x(\delta_{i}(t))\geq\beta(t,T)x(t). \end{eqnarray*}

Lemma 2.2. Assume that conditions \((H_{1})-(H_{3})\) and (1) are satisfied. If equation (1) has a nonoscillatory solution \(y\) on \([t_{0},\infty)_{\mathbb{T}}\) and \(x\) is defined as in (2) then there exists a \(T > t_{0}\) such that \begin{eqnarray*} (r(t)(x^{\Delta}(t))^{\gamma})^{\Delta}+Q(t)x^{\alpha}(t)\leq 0\ \ for \ \ t\geq T, \end{eqnarray*} where \begin{eqnarray*} Q(t)&=&q_{1}(t)\bigg(\frac{1}{m(\delta_{1}(t))}\bigg[1-\frac{p(\delta_{1}(t))}{m(\tau(\delta_{1}(t)))}\bigg]\bigg)^ {\alpha}\beta^{\alpha}(t,T)\\&&+q_{2}(t)\bigg(\frac{1}{m(\delta_{2}(t))}\bigg[1-\frac{p(\delta_{2}(t))} {m(\tau(\delta_{2}(t)))}\bigg]\bigg)^{\beta} \beta^{\beta}(t,T)>0. \end{eqnarray*}

Proof. If \(y(t)\) is an eventually positive solution of (1) then there exists a \(T\in [t_{0},\infty)_{\mathbb{T}}\) such that \begin{equation*} y(t)>0,\hspace{.2cm}y(\tau(t))>0,\hspace{.2cm}y(\delta_{i}(t))>0,\hspace{.2cm}y(\tau(\tau(t)))>0 \ \ for \hspace{.2cm}t\geq T,\hspace{.2cm}i=1,2. \end{equation*}

For \(t\geq t_{2}=\delta_{i}^{-1}(T)\) with \(i=1,2\), noting that (5), we get
From Lemma 2.1 and (6), we have \begin{eqnarray*} (r(t)(x^{\Delta}(t))^{\gamma})^{\Delta}&\leq& -q_{1}(t)y^{\alpha}(\delta_{1}(t))-q_{2}(t)y^{\beta}(\delta_{2}(t))\\ &\leq& \nonumber -q_{1}(t)\bigg(\frac{1}{m(\delta_{1}(t))}\bigg[1-\frac{p(\delta_{1}(t))}{m(\tau(\delta_{1}(t)))}\bigg]\bigg) ^{\alpha}x^{\alpha}(\delta_{1}(t))\\&& -q_{2}(t)\bigg(\frac{1}{m(\delta_{2}(t))}\bigg[1-\frac{p(\delta_{2}(t))}{m(\tau(\delta_{2}(t)))}\bigg]\bigg)^{\beta}x^{\beta} (\delta_{2}(t))\\ &\leq&\nonumber-q_{1}(t)\bigg(\frac{1}{m(\delta_{1}(t))}\bigg[1-\frac{p(\delta_{1}(t))}{m(\tau(\delta_{1}(t)))}\bigg] \bigg)^{\alpha}\beta^{\alpha}(t,T)x^{\alpha}(t)\\&& -q_{2}(t)\bigg(\frac{1}{m(\delta_{2}(t))}\bigg[1-\frac{p(\delta_{2}(t))}{m(\tau(\delta_{2}(t)))}\bigg]\bigg)^{\beta} \beta^{\beta}(t,T)x^{\beta}(t) \\ &=&\nonumber\bigg[-q_{1}(t)\bigg(\frac{1}{m(\delta_{1}(t))}[1-\frac{p(\delta_{1}(t))}{m(\tau(\delta_{1}(t)))} \bigg]\bigg)^{\alpha}\beta^{\alpha}(t,T)\\ && -q_{2}(t)\bigg(\frac{1}{m(\delta_{2}(t))}\bigg[1-\frac{p(\delta_{2}(t))}{m(\tau(\delta_{2}(t)))}\bigg]\bigg)^{\beta} \beta^{\beta}(t,T)]x^{\alpha}(t). \end{eqnarray*} Thus,

Lemma Let \(g(u)=Bu-Au^{\frac{\gamma+1}{\gamma}}\), where \(A>0\) and B are constants, \(\gamma\) is a quotient of odd positive integers. Then g attains its maximum value on \(\mathbb{R}\ \ at\ \ u^{*}=\bigg(\frac{B\gamma}{A(\gamma+1)}\bigg)^{\gamma}\) and

3. Main Results

In this section, we state and prove the main oscillation results for the equations (1).

Theroem 3.1. Assume that \((H_{1})-(H_{3})\) and (1) are satisfied. If

then every solution of (1) oscillates.

Proof. Assume the contrary and let \(y\) be a nonoscillatory solution of (1). Without loss of generality, we may assume that

We define where \begin{eqnarray*} x^{[1]}(t):=(r(t)(x^{\Delta}(t))^{\gamma})(t)\quad and \quad x^{[2]}(t):=(x^{[1]}(t))^{\Delta}. \end{eqnarray*} Then, \(w(t)>0 \ \ for \ \ t\geq T\). Since Lemma 2.1 and (2), there exists a \(T\geq t_{0}\) such that From Lemma 2.2, we get By the Ptzsche chain rule, if \(x^{\Delta}(t)>0 \ \ and \ \ \gamma>1\), then we get \begin{eqnarray*} (x^{\gamma}(t))^{\Delta}&=& \gamma \int_{0}^{1} [x(t)+\mu(t)hx^{\Delta}(t)]^{\gamma -1}x^{\Delta}(t)dh \\ &=&\gamma \int_{0}^{1} [(1-h)x(t)+hx^{\sigma}(t)]^{\gamma-1} x^{\Delta}(t) dh \\ &\geq&\gamma \int_{0}^{1} (x(t))^{\gamma-1} x^{\Delta}(t)dh \\ &=& \gamma (x(t))^{\gamma-1} x^{\Delta}(t). \end{eqnarray*} Again by the Ptzsche chain rule, if \(x^{\Delta}(t)>0 \ \ and \ \ 0< \gamma\leq 1\), then we have \begin{eqnarray*} (x^{\gamma}(t))^{\Delta}&=& \gamma \int_{0}^{1} [x(t)+\mu(t)hx^{\Delta}(t)]^{\gamma -1}x^{\Delta}(t)dh \\ &=&\nonumber \gamma \int_{0}^{1} [(1-h)x(t)+hx^{\sigma}(t)]^{\gamma-1} x^{\Delta}(t) dh \\ &\geq&\gamma \int_{0}^{1} (x^{\sigma}(t))^{\gamma-1} x^{\Delta}(t)dh \\ &=& \gamma (x^{\sigma}(t))^{\gamma-1} x^{\Delta}(t). \end{eqnarray*} Since \(x(t)\) is increasing and \(x^{[1]}(t)\) is decreasing, for \(\gamma>1\), we get Also for \(0< \gamma \leq 1\), we have Together (14) and (15), we obtain Substituting (16) in (13), we get Since \(x^{\sigma}(t)=x(t)+\mu(t)x^{\Delta}(t)\), we have Since \(x^{[1]}(t)\) is decreasing, we get \begin{eqnarray*} x(t)&=&\nonumber x(T)+\int_{T}^{t}(x^{[1]}(s))^{\frac{1}{\gamma}}\bigg(\frac{1}{r(s)}\bigg)^{\frac{1}{\gamma}} \Delta s \\ &>&(x^{[1]}(t))^{\frac{1}{\gamma}}\int_{T}^{t}\bigg(\frac{1}{r(s)}\bigg)^{\frac{1}{\gamma}} \Delta s. \end{eqnarray*} Therefore, Thus we have \begin{eqnarray*} \frac{x^{\sigma}(t)}{x(t)}=1+\frac{\mu(t)}{r^{\frac{1}{\gamma}}(t)} \frac{(x^{[1]}(t))^{\frac{1}{\gamma}}}{x(t)}\leq \frac{r^{\frac{1}{\gamma}}(t)R(t,T)+ \mu(t)}{r^{\frac{1}{\gamma}}(t)R(t,T)} \ \ for \ \ t\geq T \end{eqnarray*} and so \begin{eqnarray*} \frac{x(t)}{x^{\sigma}(t)}\geq\frac{r^{\frac{1}{\gamma}}(t)R(t,T)}{r^{\frac{1}{\gamma}}(t)R(t,T)+ \sigma(t)-t} \ \ for \ \ t\geq T. \end{eqnarray*} Thus, for \(t\geq T\), we get Consider following cases.
Case 1. Let \(\alpha < \gamma \). Since \(x^{[1]}(t)\) is positive and decreasing, it follows from Lemma 2.1 that \(x^{[1]}(t)\leq x^{[1]}(T)=c \ \ for \ \ t\geq T\). So, This implies that, Since \(\alpha < \gamma\), we get where Case 2. Let \(\alpha = \gamma \). So, \((x^{\sigma}(t))^{\alpha-\gamma}=1.\)
Case 3. Let \(\alpha > \gamma\). In this case, since \(x^{\Delta}(t)>0\), there exist \(t_{2}\geq t_{1}\) such that \(x^{\sigma}(t)>x(t)>c>0.\) This implies that \((x^{\sigma}(t))^{\alpha-\gamma}>c_{1},\) where \(c_{1}=c^{\alpha-\gamma}.\)
Combining these three cases and using the definition of \(\eta^{\sigma}\), we conclude that \begin{eqnarray*} (x^{\sigma}(t))^{\alpha-\gamma}\geq\eta^{\sigma}(t). \end{eqnarray*} Hence, Substituting (25) to (17), we obtain, \begin{eqnarray*} w^{\Delta}(t)\leq -Q(t)\bigg(\frac{r^{\frac{1}{\gamma}}(t)R(t,T)}{r^{\frac{1}{\gamma}}(t)R(t,T)+ \mu(t)}\bigg)^{\alpha}\eta^{\sigma}(t)-\gamma\frac{1} {r^{\frac{1}{\gamma}}(t)}(w^{\sigma}(t))^{\frac{1}{\gamma}+1}, \end{eqnarray*} and so It follows from the definition of \(x^{[1]}(t)\) that Integrating (28) from \(T\) to \(t\), we obtain Taking into account that \(x^{[1]}(t)\) is positive and decreasing, we get \begin{eqnarray*} x(t)\geq x(T)+ (x^{[1]}(t))^{\frac{1}{\gamma}}\int_{T}^{t} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s \ \ for \ \ t\geq T. \end{eqnarray*} Thus, we have \begin{eqnarray*} \frac{(x^{[1]}(t))^{\frac{1}{\gamma}}}{x(t)}\leq \bigg(\int_{T}^{t} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s\bigg)^{-1} \end{eqnarray*} and \begin{eqnarray*} w(t)=\frac{x^{[1]}(t)}{x^{\gamma}(t)}\leq \bigg(\int_{T}^{t} \frac{1}{r^{\frac{1}{\gamma}}(s)} \Delta s\bigg)^{-\gamma} \ \ for \ \ t\in[T,\infty)_{\mathbb{T}}, \end{eqnarray*} which implies, in view of \((H_{2})\), that \begin{eqnarray*} \displaystyle\lim_{t\rightarrow\infty}w(t)=0. \end{eqnarray*} Integrating (27) from \(T\) to \(\infty\) and using the fact that \(\lim_{t\rightarrow\infty}w(t)=0\) we obtain \begin{eqnarray*} w(T)\geq \int_{T}^{\infty} Q_{1}(s) \Delta s. \end{eqnarray*} which contradicts (9). The proof is completed.

Theroem 3.2. Assume that \((H_{1})-(H_{3})\) and (1) are satisfied. If there exists a positive rd-continuous \(\Delta \)-differentiable function \(z(t)\) such that

\noindent then every solution of (1) oscillates.

Proof. Assume the contrary and let \(y\) be a nonoscillatory solution of (1). Without loss of generality, we may assume that

Let \(w\) be defined as in (11), then \(w(t)>0 \ \ \textrm{for} \ \ t\geq T \). Using (26), the following inequality is true: Multiplying (32) by \(z(t)\) and integrating from \(T\) to \(t\), we obtain Integration by parts, we get \begin{eqnarray*} -\int_{T}^{t} z(s)w^{\Delta}(s)\Delta s&=&-z(t)w(t)+z(T)w(T)+ \int_{T}^{t}z^{\Delta}(s)w^{\sigma}(s) \Delta s \\ &\leq&z(T)w(T)+\int_{T}^{t}z^{\Delta}(s)w^{\sigma}(s) \Delta s. \end{eqnarray*} It follows that \begin{eqnarray*} \int_{T}^{t} z(s)Q_{1}(s)\leq z(T)w(T)+\int_{T}^{t}z^{\Delta}(s)w^{\sigma}(s) \Delta s -\int_{T}^{t}\frac{\gamma z(s)}{r^{\frac{1}{\gamma}}(s)}(w^{\sigma}(s))^{\frac{1}{\gamma}+1} \Delta s. \end{eqnarray*} Setting \(B=z^{\Delta}(s),A=\frac{\gamma z(s)}{r^{\frac{1}{\gamma}}(s)}, \ \ u=w^{\sigma}(s)\) and using Lemma 2.3, we get \begin{eqnarray*} \int_{T}^{t}z(s)Q_{1}(s) \Delta s \leq z(T)w(T)+\int_{T}^{t}\frac{\gamma^{\gamma}}{(\gamma+1)^{\gamma+1}} \frac{(z^{\Delta}(s))^{\gamma+1}r(s)}{\gamma^{\gamma}z^{\gamma}(s)} \end{eqnarray*} and so which contradicts condition (30). Then every solution of (1) oscillates. The proof is completed.

Theorem 3.3. Assume that \((H_{1})-(H_{3})\) and (1) are satisfied. Suppose that \(z(t)\) is defined as in Theorem 3.2, \(H \in\mathbb{R}\) and for \(t>s\),

is satisfied. Then every solution of (1) oscillates.

Proof. Assume the contrary and let \(y\) be a nonoscillatory solution of (1). Without loss of generality, we may assume that

Let \(w\) be defined as in (11), then \(w(t)>0 \ \ for \ \ t\geq T\). Multiplying (32) by \(z(t)H(t,u)\) and integrating from \(T\) to \(t\), we obtain for \(u=t\). Integration by parts we get \begin{eqnarray*} -\int_{T}^{t} H(t,s)z(s)w^{\Delta}(s)\Delta s = H(t,T)z(T)w(T)+ \int_{T}^{t}(H(t,s)z(s))^{\Delta}w^{\sigma}(s) \Delta s. \end{eqnarray*} Thus, we have \begin{eqnarray*} &&\int_{T}^{t}H(t,s) z(s)Q_{1}(s) \Delta s\\&\leq&\nonumber H(t,T)z(T)w(T)+\int_{T}^{t}(H^{\Delta}(t,s)z(s)+H^{\sigma}(t,s) z^{\Delta}(s))w^{\sigma}(s) \Delta s \\ && - \int_{T}^{t}\frac{\gamma z(s)H(t,s)} {r^{\frac{1}{\gamma}}(s)}(w^{\sigma}(s))^{\frac{1}{\gamma}+1} \Delta s \\ &=&\nonumber H(t,T)z(T)w(T)+\int_{T}^{t}C(t,s)w^{\sigma}(s) \Delta s -\int_{T}^{t}\frac{\gamma z(s)H(t,s)} {r^{\frac{1}{\gamma}}(s)}(w^{\sigma}(s))^{\frac{\gamma+1}{\gamma}} \Delta s. \end{eqnarray*} Setting \(B=C(t,s),\quad A=\frac{H(t,s)\gamma z(s)}{r^{\frac{1}{\gamma}}(s)}\), \(u=w^{\sigma}(s)\) and using Lemma 2.3, we have \begin{eqnarray*} \int_{T}^{t}H(t,s) z(s)Q_{1}(s) \Delta s \leq H(t,T)z(T)w(T)+\int_{T}^{t}\frac{\gamma^{\gamma}}{(\gamma+1)^{\gamma+1}} \frac{C^{\gamma+1}(t,s)r(s)}{H^{\gamma}(t,s)z^{\gamma}(s)\gamma^{\gamma}} \Delta s \end{eqnarray*} and which contradicts condition (30). Then every solution of (1) oscillates. The proof is completed.

4. Example

Example 4.1. Let \(\mathbb{T}\) be any time scales and we consider the following second order neutral dynamic equation

where \(t\in[2,\infty)_{\mathbb{T}}.\)
\((H_{1}) \ \ \tau(t)=\frac{t}{2}\leq t,\quad \delta_{1}(t)=t-\frac{1}{2}< t,\quad \delta_{2}(t)=t-\frac{1}{4}< t, \displaystyle\lim_{t\rightarrow\infty}\tau(t)=\displaystyle\lim_{t\rightarrow\infty}\delta_{i}(t)=\infty,\quad i=1,2 \).
\((H_{2})\ \ \tau(t)=\frac{t}{2}\in C_{rd}(\mathbb{T},\mathbb{R}^{+})\), \(r(t)=1 \in C_{rd}(\mathbb{T},\mathbb{R}^{+})\) that \(\int_{2}^{\infty}\frac{1}{r^{\frac{1}{\gamma}}(t)}\Delta t=\int_{2}^{\infty}\Delta t= \lim_{a\rightarrow\infty}\int_{2}^{a}\Delta t=\displaystyle\lim_{a\rightarrow\infty}a-2=\infty\), \(m(t)=t\in C_{rd}(\mathbb{T},\mathbb{R}^{+})\), \(p(t)=\frac{1}{2}\in C_{rd}(\mathbb{T},[0,1))\) that \(m(\tau(t))=\frac{t}{2}\geq1>\frac{1}{2}=p(t)\) for \(t\geq2\).
\((H_{3})\ \ q_{1}(t)=\frac{(\sigma(t))^{\frac{3}{5}}}{(2t-3)^{\frac{1}{5}}},\quad\alpha=\frac{1}{5},\quad q_{2}(t)=\frac{(\sigma(t))^{\frac{3}{5}}}{(4t-5)^{\frac{1}{5}}},\quad \beta=\frac{1}{5},\quad \textrm{that} \quad q_{i}(t)\in C_{rd}(\mathbb{T},\mathbb{R}^{+})\).
Since \begin{eqnarray*} Q(t)&=&q_{1}(t)\bigg(\frac{1}{m(\delta_{1}(t))}\bigg[1-\frac{p(\delta_{1}(t)}{m(\tau(\delta_{1}(t)))}\bigg]\bigg) ^{\alpha}\beta^{\alpha}(t,T)\\&&+q_{2}(t)\bigg(\frac{1}{m(\delta_{2}(t))}\bigg[1-\frac{p(\delta_{2}(t)} {m(\tau(\delta_{2}(t)))}\bigg]\bigg)^{\beta}\beta^{\beta}(t,T)\\ &=&\frac{(\sigma(t))^{\frac{3}{5}}}{(2t-3)^{\frac{1}{5}}}\bigg(\frac{1}{t-\frac{1}{2}} \bigg[1-\frac{1}{2.\frac{1}{2}(t-\frac{1}{2})}\bigg]\bigg)^{\frac{1}{5}}\bigg(\frac{\delta_{1}(t)-T}{t-T}\bigg) ^{\frac{1}{5}}\\&&+\frac{(\sigma(t))^{\frac{3}{5}}}{(4t-5)^{\frac{1}{5}}}\bigg(\frac{1}{t-\frac{1}{4}} \bigg[1-\frac{1}{2.\frac{1}{2}(t-\frac{1}{4})}\bigg]\bigg)^{\frac{1}{5}}\bigg(\frac{\delta_{2}(t)-T}{t-T}\bigg) ^{\frac{1}{5}}\\ &=&\frac{(\sigma(t))^{\frac{3}{5}}}{(2t-3)^{\frac{1}{5}}}\frac{2^{\frac{1}{5}}}{(2t-1)^{\frac{1}{5}}} \frac{(2t-3)^{\frac{1}{5}}}{(2t-1)^{\frac{1}{5}}}\bigg(\frac{\delta_{1}(t)-T}{t-T}\bigg) ^{\frac{1}{5}}\\&&+\frac{(\sigma(t))^{\frac{3}{5}}}{(4t-5)^{\frac{1}{5}}}\frac{4^{\frac{1}{5}}}{(4t-1)^{\frac{1}{5}}} \frac{(4t-5)^{\frac{1}{5}}}{(4t-1)^{\frac{1}{5}}}\bigg(\frac{\delta_{2}(t)-T}{t-T}\bigg) ^{\frac{1}{5}}, \end{eqnarray*} we get \begin{equation*} Q_{1}(t)= Q(t)\bigg(\frac{r^{\frac{1}{\gamma}}(t)R(t,T)}{r^{\frac{1}{\gamma}}(t)R(t,T)+\mu(t)}\bigg)^{\alpha}\eta^{\sigma}(t), \end{equation*} where \begin{eqnarray*} \bigg(\frac{r^{\frac{1}{\gamma}}(t)R(t,T)}{r^{\frac{1}{\gamma}}(t)R(t,T)+\mu(t)}\bigg)^{\alpha}= \bigg(\frac{t-T}{t-T+\sigma(t)-t}\bigg)^{\frac{1}{5}}=\bigg(\frac{t-T}{\sigma(t)-T}\bigg)^{\frac{1}{5}}. \end{eqnarray*} and \begin{eqnarray*} \eta^{\sigma}(t)=c_{2}^{\alpha-\gamma}\bigg(\int_{T}^{\sigma(t)}\frac{1}{r^{\frac{1}{\gamma}}(s)}\Delta s\bigg)^{\alpha-\gamma}=c_{2}^{-\frac{2}{5}}(\sigma(t)-T)^{-\frac{2}{5}}. \end{eqnarray*} Thus we get \begin{eqnarray*} Q(t)\bigg(\frac{t-T}{\sigma(t)-T}\bigg)^{\frac{1}{5}}&=&\frac{(\sigma(t))^{\frac{3}{5}}2^{\frac{1}{5}}} {(2t-1)^{\frac{2}{5}}}\bigg(\frac{\delta_{1}(t)-T}{t-T}\bigg) ^{\frac{1}{5}}\bigg(\frac{t-T}{\sigma(t)-T}\bigg)^{\frac{1}{5}}\\&&+\frac{(\sigma(t))^{\frac{3}{5}}4^{\frac{1}{5}}} {(4t-1)^{\frac{2}{5}}}\bigg(\frac{\delta_{2}(t)-T}{t-T}\bigg) ^{\frac{1}{5}}\bigg(\frac{t-T}{\sigma(t)-T}\bigg)^{\frac{1}{5}}\\&>& \frac{(\sigma(t))^{\frac{3}{5}}2^{\frac{1}{5}}} {2^{\frac{2}{5}}t^{\frac{2}{5}}}\bigg(\frac{\delta_{1}(t)-T}{\sigma(t)-T}\bigg)^{\frac{1}{5}}+ \frac{(\sigma(t))^{\frac{3}{5}}4^{\frac{1}{5}}} {4^{\frac{2}{5}}t^{\frac{2}{5}}}\bigg(\frac{\delta_{2}(t)-T}{\sigma(t)-T}\bigg)^{\frac{1}{5}}. \end{eqnarray*} Archimedes property says if \(x\) and \(y\) are real numbers with \(x>0\), there exists a natural \(n\) such that \(nx>y\). So, for \(t>T\) and \(\delta_{i}(t)>T\) there exist constants \(k_{1}, k_{2}>0\) sufficiently large that \begin{eqnarray*} Q_{1}(t)&>&\frac{1}{c_{2}^{\frac{2}{5}}(\sigma(t)-T)^{\frac{2}{5}}}\bigg\{\frac{(\sigma(t))^{\frac{3}{5}}2^{\frac{1}{5}}} {2^{\frac{1}{5}}t^{\frac{2}{5}}}\bigg(\frac{t}{k_{1}\sigma(t)}\bigg)^{\frac{1}{5}} +\frac{(\sigma(t))^{\frac{3}{5}}4^{\frac{1}{5}}} {4^{\frac{1}{5}}t^{\frac{2}{5}}}\bigg(\frac{t}{k_{2}\sigma(t)}\bigg)^{\frac{1}{5}}\bigg\}\\ &>&\frac{(\sigma(t))^{\frac{3}{5}}2^{\frac{1}{5}}} {2^{\frac{1}{5}}t^{\frac{2}{5}}}\bigg(\frac{t}{k_{1}\sigma(t)}\bigg)^{\frac{1}{5}} \frac{1}{c_{2}^{\frac{2}{5}}(\sigma(t))^{\frac{2}{5}}} +\frac{(\sigma(t))^{\frac{3}{5}}4^{\frac{1}{5}}} {4^{\frac{1}{5}}t^{\frac{2}{5}}}\bigg(\frac{t}{k_{2}\sigma(t)}\bigg)^{\frac{1}{5}} \frac{1}{c_{2}^{\frac{2}{5}}(\sigma(t))^{\frac{2}{5}}}\\ &=&\frac{1}{2^{\frac{1}{5}}k_{1}^{\frac{1}{5}}c_{2}^{\frac{2}{5}}t^{\frac{1}{5}}} +\frac{1}{4^{\frac{1}{5}}k_{2}^{\frac{1}{5}}c_{2}^{\frac{2}{5}}t^{\frac{1}{5}}}. \end{eqnarray*} Let \(z(t)=1\), thus we obtain, \begin{eqnarray*} &&\displaystyle\lim_{t\rightarrow\infty}\sup\int_{T}^{t} \bigg[z(s)Q_{1}(s)-\frac{r(s)(z^{\Delta}(s))^{\gamma+1}} {(\gamma+1)^{\gamma+1}z^{\gamma}(s)}\bigg]\\&\geq&\displaystyle\lim_{t\rightarrow\infty}\sup\int_{T}^{t} \bigg[\frac{1}{2^{\frac{1}{5}}k_{1}^{\frac{1}{5}}c_{2}^{\frac{2}{5}}} +\frac{1}{4^{\frac{1}{5}}k_{2}^{\frac{1}{5}}c_{2}^{\frac{2}{5}}}\bigg]\frac{1}{s^{\frac{1}{5}}}\Delta s=\infty. \end{eqnarray*}

According to Theorem 3.2, every solution of (39) is oscillatory.

Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Let \(G=(V,E)\) be a graph, where \(V(G)\) and \(E(G)\) are denoted as the vertex set and the edge set of \(G\), respectively. All graphs considered in this paper are finite, loopless, and without multiple edges. Notations and terminologies used but undefined in this paper can be found in Bondy and Mutry [1].

Graph coloring can be regarded as a special case of graph labeling, and it is an assignment of labels called ``colors'' to elements of a graph subject to certain constraints. Specifically, we present some examples below:
\(\bullet\) vertex coloring (or, called proper vertex coloring), assigns colors to each vertex of a graph so that no two adjacent vertices will be assigned with the same color. A vertex coloring of a graph with \(k\) or fewer colors is known as a \(k\)-coloring, and this graph is said to be a \(k\)-colorable graph. The chromatic number of a graph \(G\) (denoted by \(\chi(G)\)) is the smallest value of \(k\) possible to obtain a \(k\)-coloring.
\(\bullet\) total coloring, assigning colors to each vertex and each edge of a graph so that no adjacent elements are assigned the same color. The total chromatic number \(\chi''(G)\) of a graph \(G\) is the smallest number of colors needed in the total coloring of \(G\).
\(\bullet\) harmonious coloring, is a proper vertex coloring of a graph \(G\) that each pair of colors appears on at most one edge. The harmonious chromatic number of \(G\), \(\chi_{h}(G)\), is the minimum number of colors needed for the harmonious coloring of \(G\).
\(\bullet\) fractional coloring, is a branch of fractional theory (see Scheinesrman and Ullman [2] for more details). The \(a:b\) coloring of a graph, i.e., assigns each vertex \(v\) a set \(C_{v}\in\{1,\cdots,a\}\) meets \(|C_{v}|=b\), and \(C_{v}\cap C_{v'}=\emptyset\) if \(vv'\in E(G)\). Then. such \(a:b\) coloring is just the fractional color of graph \(G\). The corresponding fractional chromatic number \(\chi_{f}(G)\) is defined by $$\chi_{f}(G)=\inf\{\frac{a}{b}| G\quad {\rm exists\quad a}\quad a:b\quad {\rm coloring}\}.$$ \(\bullet\) fractional total coloring, is a combination of fractional coloring and total coloring (see Kilakos and Reed [3] for more details). We say that a graph \(G\) is \(\frac{a}{b}\)-fractional total colorable if there exist a fractional \(\frac{a}{b}\)-coloring of the total graph of \(G\). The fractional total chromatic number \(\chi_{f}^{''}(G)\) of a graph \(G\) is the smallest number of fractional coloring needed in its total graph.
\(\bullet\) \(n'\)-path distinguishing coloring, see Harary [4] for more details, is a class of proper coloring in which all vertices are colored different in each path \(P_{n'}\) of length \(n'\). The corresponding \(n'\)-path distinguishing chromatic number \(\chi_{P_{n'}}^{''}(G)\) of a graph \(G\) is the smallest number of \(n'\)-path distinguishing coloring needed in \(G\).

Harmonious coloring, as an important topic of graph coloring, has raised much attention among the researchers. Hopcroft and Krisnamoorthy [5] first introduced the concept of harmonious coloring, and showed that the harmonious coloring problem for general graphs is NP-complete. Lee and Mitchum [6] presented an upper bound for the harmonious chromatic number of a graph. Miller and Pritikin [7] constructed efficient harmonious colorings of complete binary trees, 2 and 3-dimensional grids, and \(n\)-dimensional cubes. Ioannidou and Nikolopoulos [8] studied the harmonious coloring on subclasses of colinear graphs. Hegde and Castelino [9] investigated the proper harmonious coloring number of graphs such as alternating paths and alternating cycles. Venkatachalam et al. [10] reported the harmonious chromatic number for the central graph, middle graph, total graph and line graph of double star graph \(K_{1,n,n}\), and they proved that for the line graph of double star graph, the harmonious chromatic number and the achromatic number are equal. Furthermore, this result can be extended by classifying the different families of graphs for which these two numbers are equal. Akbari et al. [12] obtained the harmonious coloring of trees with large maximum degree \(\ge\frac{n+2}{3}\). Edwards [13] gave an upper bound for the harmonious chromatic number of a general directed graph, and showed that determining the exact value of the harmonious chromatic number is NP-hard for directed graphs of bounded degree. Muntaner-Batle et al. [14] found the harmonious chromatic number of the corona product of any graph $G$ of order $l$ with the complete graph \(K_{n}\) for \(l\le n\). As a consequence of this work, then also obtained the harmonious chromatic number of \(t\) copies of \(K_{n}\) for \(t\le n+1\). Hegde and Castelino [14] investigated the proper harmonious coloring number of graphs such as unidirectional paths, unicycles, inspoken and outspoken wheels, \(n\)-ary trees of different levels etc.

However, the standard harmonious coloring is much difficult than any other kind of graph coloring. From this point of view, a relax version of harmonious coloring was introduced as locally harmonious coloring. The locally harmonious coloring is a kind of proper vertex coloring which only needs that the adjacent edges have different color pairs, i.e., for each vertex \(v\), its adjacent vertices are all colored different (no two vertices in \(N(v)\) colored the same). The locally harmonious chromatic number of \(G\) denoted by \(\chi_{LH}(G)\) is the smallest number \(k\) that \(G\) admits a \(k\)-locally harmonious coloring.

In this paper, we further consider the locally harmonious coloring, and introduce an extension coloring concept. The fractional locally harmonious coloring is the rational approach to locally harmonious coloring so that each vertex \(v\) in a graph \(G\) assigns a collection with \(b\) element from \(\{1,\cdots,a\}\) denoted by \(C_{v}\), we have \(C_{v}\cap C_{v'}=\emptyset\) if \(vv'\in E(G)\), and \(C_{v'}\cap C_{v''}=\emptyset\) if \(v'\) and \(v''\) adjacent to the same vertex \(v\). The fractional locally harmonious chromatic number of \(G\) denoted by \(\chi_{FLH}(G)\) is the smallest rational number \(\frac{a}{b}\) so that \(G\) admits a \(\frac{a}{b}\)-fractional locally harmonious coloring. Obviously, we have \(\chi_{FLH}(G)\le \chi_{LH}(G)\) for any graph \(G\) and the fractional locally harmonious coloring problem for general graphs is also NP-complete.

2. Main Results and Proofs

In this section, we aim to present our main conclusions.

Theorem 2.1. Let \(D(G)\) be the diameter of graph \(G\). Then, \(\chi_{FLH}(G)=|V(G)|\Leftrightarrow D(G)\le2\).

Proof. For the necessity. If \(\chi_{FLH}(G)=|V(G)|\) but \(D(G)\ge3\), then there exist two vertices \(u,v\in V(G)\) satisfying \(d(u,v)\ge3\). Thus, by assigning the same color to \(u\) and \(v\), we infer that \(\chi_{LH}(G)\le|V(G)|-1< |V(G)|\). Hence, \(\chi_{FLH}(G)\le\chi_{LH}(G)< |V(G)| \), a contradiction.

For the sufficiency, we assume that \(D(G)\le2\) and \(C\) is a fractional locally harmonious coloring of \(G\). Let \(C_{v}\) be the collection of elements assigned to vertex \(v\).
\(\bullet\) If \(D(G)=1\), then \(G\cong K_{|V(G)|}\). Clearly, we have \(\chi_{FLH}(G)=|V(G)|\).
\(\bullet\) If \(D(G)=2\), then there are two situations for any two vertices \(u,v\in V(G)\). 1) If \(uv\in E(G)\), the \(C_{u}\cap C_{v}=\emptyset\). 2) If \(uv\notin E(G)\), by means of \(D(G)=2\), there exists a vertex \(w\) such that \(uw\in E(G)\) and \(vw\in E(G)\). Using the definition of fractional locally harmonious coloring, we infer that \(C_{u}\cap C_{v}=\emptyset\).

From the above discussion, we summarize that \(C_{u}\cap C_{v}=\emptyset\) for any two vertices \(u,v\in V(G)\). Therefore, \(\chi_{FLH}(G)=|V(G)|\).

Now, we extend the \(n'\)-path distinguishing coloring to its fractional version. Assign each vertex \(v\) a set \(C_{v}\in\{1,\cdots,a\}\) such that \(|C_{v}|=b\), and \(C_{v}\cap C_{v'}=\emptyset\) if \(vv'\in E(G)\). The \(a:b\) fractional \(n'\)-path distinguishing coloring is defined as the fractional coloring of graph \(G\) such that \(C_{v}\cap C_{v'}=\emptyset\) for any two vertices \(v,v'\in P_{n'}\) with \(P_{n'}=n'\). The corresponding fractional \(n'\)-path distinguishing chromatic number \(\chi_{FP_{n'}}(G)\) is defined as mimimum \(\frac{a}{b}\) such that \(G\) has \(\frac{a}{b}\)-fractional \(n'\)-path distinguishing coloring. The second main result in our remark is stated as follows which presented the equivalence between \(\frac{a}{b}\)-fractional locally harmonious coloring and \(\frac{a}{b}\)-fractional \(3\)-path distinguishing coloring.

Theorem 2.2. A graph \(G\) can be \(\frac{a}{b}\)-fractional locally harmonious coloring if and only if it has \(\frac{a}{b}\)-fractional \(3\)-path distinguishing coloring, i.e., $$\chi_{FLH}(G)=\frac{a}{b}\Leftrightarrow \chi_{FP_{3}}(G)=\frac{a}{b}.$$

Proof. For the necessity. Since \(G\) is \(\frac{a}{b}\)-fractional locally harmonious colorable, we infer that \(C_{v_{1}}\cap C_{v_{2}}=\emptyset\) for each \(v\in V(G)\) and \(v_{1},v_{2}\in N[v]\). Hence, for any 3-path \(P_{3}\triangleq v_{1}v_{2}v_{3}\), we have \(v_{1},v_{3}\in N[v_{2}]\), and the intersection of assigned collection of \(v_{1},v_{2},v_{3}\) is \(\emptyset\), i.e., \(G\) is \(\frac{a}{b}\)-fractional \(3\)-path distinguishing colorable.

For the sufficiency. Assume that \(G\) is \(\frac{a}{b}\)-fractional \(3\)-path distinguishing colorable, but not a \(\frac{a}{b}\)-fractional locally harmonious coloring graph. Then, there exist a vertex \(v\in V(G)\), and \(v_{1},v_{2}\in N[v]\) such that \(C_{v_{1}}\cap C_{v_{2}}\ne\emptyset\). This contracts that \(v_{1},v,v_{2}\) in a \(P_{3}\) path.

For the relationship between cut vertex and fractional locally harmonious coloring, we present the following conclusion.

Theorem 2.3. Let \(u\) be a cut vertex of graph \(G\), and \(G_{i}^{'} (i=1,\cdots,t)\) be the branches of \(G-\{u\}\). Let \(N_{G}[u]=N_{G}(u)\cup\{u\}\). Set $$G_{i}=G[V(G_{i}^{'})\cup N_{G}[u]]$$ for \(i=1,\cdots,t\). Then, we have $$\chi_{FLH}(G)=\max\{\chi_{FLH}(G_{i}),i=1,\cdots,t\}\triangleq\frac{a}{b}.$$

Proof. Let \(i_{i}=\{i|\chi_{FLH}(G_{i})=\frac{a}{b},i=1,\cdots,t\}\). First, we \(\frac{a}{b}\)-fractional locally harmonious coloring one of \(G_{i_{0}}\) which denoted by \(C_{1}\). Then, \(\frac{a}{b}\)-fractional locally harmonious coloring the rest \(G_{i}\) one by one so that the collection assigned for the vertices in \(N_{G}[u]\) is the same with the collection assigned to these vertices under \(C_{1}\), and such colorings are denoted by \(C_{i} (i=2,\cdots,t)\) which exist obviously. Finally, set \(C=\cup_{i=1}^{t}C_{i}\). Thus, we get the desired \(\frac{a}{b}\)-fractional locally harmonious coloring.

The following conclusion reveals the fractional locally harmonious chromatic number of cycle, and we skip the detail proof.

Theorem 2.4. Let \(C_{n}\) be a cycle of order \(n\). Then, $$\chi_{FLH}(C_{n})=\left\{\begin{array}{ll}n,& \hbox{if $n\le5$} \\ \frac{n}{k},& \hbox{if $n\ge6$ and $k=\lfloor\frac{n}{3}\rfloor$}. \end{array}\right.$$

The following corollaries are deduced immediately from Theorem 2.4.

Theorem 2.5. If \(n\equiv0\)(mod 3), then \(\chi_{FLH}(C_{n})=3\).

Corollary 2.6. If \(n\ge6\), then \(3\le \chi_{FLH}(C_{n})\le4\).

3. Conclusion Graph coloring theory is the core research contents of graph theory. It has important applications in optimization theory, task scheduling, and computer networks. In this remark, we give the new concept called fractional locally harmonious coloring, and determine several properties for this coloring.

Acknowledgments

The authors thank the reviewers for their constructive comments in improving the quality of this paper.

Funding

The research is partially supported by NSFC (nos. 11401519, 11371328, and 11471293).

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Heat conduction problems frequently occur in many industrial processes thereby necessitating much attention from researchers. Of recent, these problems tend to be modelled with fractional order derivatives in either time or space variables or both. In light of this, the study of fractional differential equations [1, 2, 3] becomes vital in this regards. Moreover, many methods have been employed by many researchers to tackle varieties of heat conduction problems ranging from analytical down to approximate methods such that the novel series method for fractional diffusion equation by Yan et al. [4], an approximate decomposition method solution for a fractional diffusion-wave equation by Al-Khaled and Momani [5], the Adomian decomposition method for a fractional diffusion equation, nonlinear heat equation, heat equation with nonlocal boundary conditions and nonlinear diffusion equations, respectively [6, 7, 8, 9, 10]. Other methods include, the symmetry method for classifications of (2+1)-nonlinear heat equation by Ahmad et al. [11], the Sumudu Homotopy Perturbation Method (SHPM) for fractional KdV equations [12], the computational approach based on ADM [13], the Laplace transform method for fractional fluid flow and oscillatory process equations [14] and lastly the Wiener-Hopf method [15, 16, 17] for semi-infinite heat problems among others.

However, in the present article, a time fractional diffusion problem is formulated with special boundary conditions, specifically, the nonlocal boundary conditions. This new problem is aimed to be solved through utilizing the well-known Laplace transform method [18] alongside employing the Adomian decomposition method [7] in what is termed as the Laplace decomposition method, [19, 20, 21, 22, 23]. Further, in order to achieve this, a modification to Beilin's lemma [24] to feature fractional derivative in time variable will be given.

The paper is organized as follows: In Section 2, we present some basics about the fractional calculus. Section 3 gives the formulation of the problem under consideration. In Section 4, we give the analysis of the methodology and in Section 5, we present some application and results, and finally, Section 6 gives the conclusion.

Fractional Calculus and Some Definitions

In this section, we give some preliminary definitions of fractional calculus theory which will be used later on as follows:

Definition 2.1. [Caputo Fractional Derivative] The Caputo derivative of a casual function \(u(t)\) \((u(t)=0, \ t < 0)\) with \(\alpha>0\) is defined by [19]

Where \(\Gamma(.)\) is the well-known gamma function defined by \begin{equation*} (x-1)!=\Gamma(x)=\int_{0}^{\infty}e^{-t} t^{x-1} dt. \end{equation*} Some useful properties of the Caputo derivative are given below:

1. \(D^{\alpha}_t t^r=\frac{\Gamma(1+r)}{\Gamma(1+r-\alpha)} t^{r-\alpha},\)
3. \(D^{\alpha}_t \left[ c u(t)\right] = c D^{\alpha}_t u(t), \ \ c\) constant,
5. \(D^{\alpha}_t c = 0,\) \(c\) constant,
7. \(D^{\alpha}_t \left[ c u(t) + k v(t) \right] = c D^{\alpha}_t \left[ u(t) \right]+ k D^{\alpha}_t \left[ v(t) \right],\)
9. \(D^{\alpha}_t \left[ u(t) v(t) \right] = v(t) D^{\alpha}_t \left[ u(t) \right] + u(t) D^{\alpha}_t \left[ v(t) \right] .\)

For more, see [1, 2, 3].

Definition 2.2. [Laplace Transform for Caputo Fractional Derivative] The Laplace transform for Caputo fractional derivative is given by

Definition 2.3. [Mittag-Leffler Function] The one parameter Mittag-Leffler function is given by [2]

Definition 2.4.[Formulation of the Problem] We consider the time-fractional 2-dimensional heat diffusion equation of the form

subject to the initial condition and the nonlocal boundary conditions Furthermore, the function \(g(x,y)\) is assumed to satisfy the comparability conditions; that is \begin{equation*} \begin{split} g(0,0)& =0,\\ \int_{0}^{1}g(x,y)dx & =0, \quad y=0 \\ \int_{0}^{1}g(x,y)dy & =0, \quad x=0. \end{split} \end{equation*} Here, we give the following lemma by virtue of the modified Beilin's lemma [23] in Caputo fractional derivative sense to transform problem (4)-(7) to an equivalent boundary value problem with classical boundary conditions

Lemma 2.5. Problem (4)-(7) is equivalent to the following problem

Proof. Let \(u(x,t,t)\) be a solution of (4)-(7). Integrating (4) w.r.t `\(x\)' and `\(y\)' over \((0,l)\) respectively alongside utilizing (6)-(7), we get

and Differentiating (9) w.r.t `\(y\)' at \(y=0\) and (10) w.r.t `\(x\)' at \(x=0\) and thereafter add them, we get Now let \(u(x,t,t)\) be a solution of problem (8), the we show the following and To show this, we integrate (4) w.r.t `\(x\)' and yields \begin{equation*} \frac{\partial}{\partial t}\int_{0}^{1}u(x,y,t)dx-\frac{\partial^2}{\partial x^2}\int_{0}^{1}u(x,y,t)dx-\frac{\partial^2}{\partial y^2}\int_{0}^{1}u(x,y,t)dx=0. \end{equation*} Thus, by virtue of the compatibility conditions, we get \begin{equation*} \int_{0}^{1}u(x,y,t)dx =0, \ \ \ \ \forall t \in (0,T).\\ \end{equation*} Similarly, \begin{equation*} \int_{0}^{1}u(x,y,t)dy =0, \ \ \ \ \forall t \in (0,T). \end{equation*}

3. Analysis of the Method

To illustrate the basic idea of the method, we consider a general nonlinear nonhomogeneous time-fractional partial differential equation with initial conditions of the following form: subject to the initial condition where \(u^\alpha_t\) is the Caputo derivative of order \(\alpha\), and \(f(x,t)\) is the source function; \(L\) represents a linear fractional differential operator and \(N\) is the general nonlinear fractional differential operator.
The method first starts by taking the Laplace transform of equation (14) in \(t\), subject to the prescribed initial conditions given in equation (15), we obtain Now, taking the inverse Laplace transform of equation (17) and attaching the nonhomogeneous term with the initial conditions, yields Now, from equation (18), we assume the unknown function \(u(x,t)\) to have the series solution and the nonlinear term \(N\left(u(x,t)\right)\) by the Adomian polynomials [7]; where \(A_m\)'s are the Adomian polynomials, see [7]. Thus, equation (18) becomes Thus we identify \(u_0 (x,t)\) with the initial condition term and the term resulting from the nonhomogeneous term; and the rest of the components \(u_m (x,t)\) are determined recursively as shown below:

4. Applications and Results

In this section, we apply the proposed method to two different time-fractional 2-dimensional heat diffusion equations and later illustrated the solutions graphically in figures 1a, 1b, 1c, 2a 2b and 2c with the aid of Mathematica software as follows:

Example 4.1. Consider the time-fractional 2-dimensional heat diffusion equation equation

with the initial condition and the boundary conditions First, we transform our system (22)-(24) using Lemma 2.5 to obtain a system solvable by the Laplace decomposition method as follows: subject to the new conditions \begin{equation*} \left\{ \begin{array}{rcl} u(x,y,0) =\sin(x)\sin(y), \\ u(0,0,t) =0, \\ u_{xy}(l,0,t) +u_{xy}(0,l,t)\\-2u_{xy}(0,0,t)=0. \\ \end{array}\right. \end{equation*} Then, on taking the Laplace transform of both sides of equation (25) subject to the initial condition, we obtain Taking the inverse Laplace transform of equation (27), we get Now, from equation (28), we assume the unknown function \(u(x,y,t)\) to have the series solution Thus, equation (28) becomes Thus we identify \(u_0 (x,y,t)\) with the initial condition term that originate from the initial condition; and the rest of the components \(u_m (x,y,t)\) are determined recursively by: We now obtain some few terms from equation (31) as follows and so on. We therefore sum up the above iterations to get which leads to the exact solution The graph of the solution of equation (37) is shown in Figure \(1a, 1b\) and \(1c\) as follows:

Example 4.2. Consider the time-fractional 2-dimensional heat diffusion equation equation

with the initial condition and the boundary conditions Proceeding as above after obtaining the solvable system with the help of {\bf{Lemma 3.1}}, we get the solutions recursively as: Some few terms from equation (41) are as follows and so on. We therefore sum up the above iterations to get which leads to the exact solution The graph of the solution of equation (47) is shown in Figure \(2a, 2b\) and \(2c\) as follows:

5. Conclusion

In conclusion, a time fractional diffusion problem is formulated with nonlocal boundary conditions. This new problem is then solved through utilizing the Laplace transform method coupled to the well-known Adomian decomposition method after employing the modified version of Beilin's lemma featuring fractional derivative in time. Some test problems are solved and presented graphically with the aid of Mathematica software.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Let \(H\) be the class of functions analytic in \(U=\{z:\lvert z\rvert< 1\}\) and let H[a,n] be the subclasses of \(H\) consisting of functions of the form \(f(z)=a+a_{n}z^{n}+a_{n+1}z^{n+1}+a_{n+2}z^{n+2}+...\, \). Let \(A\) be the subclasses of \(H\) consisting of functions of the form \(f(z)=z+a_{2}z^{2}+a_{3}z^{3}+... \) or Let \(A(n)\) denote the class of functions \(f(z)\) of the form \begin{align} a_{k+1}\geq0, n\in \{1,2,3,...\} \nonumber \end{align} which are analytic in the open unit disk \(U=\{z:\lvert z\rvert< 1\}\).
Next, we define \((n,\delta)\)-neighbourhood for the functions belonging to class \(A(n)\) and also for identity function.

Definition 1.1. [\((n,\delta)\)-neighbourhood] By following the earlier investigations by Goodman [1] and Ruscheweyh [2], for any \(f(z)\in A(n)\) and \(\delta \geq0\), we define the \((n,\delta)\)-neighbourhood of \(f\) by

In particular for the identity function \(e(z)=z\) we have We say that the function \(f(z)\in A(n)\) is said to be starlike functions of complex order \(\gamma\) or \(f(z)\in S_{n}^{\ast}(\gamma)\) if it satisfies the inequality Furthermore, a function \(f(z)\in A(n)\) is said to be convex functions of complex order \(\gamma\) or \(f(z)\in C_{n}^{\ast}(\gamma)\) if it satisfies the inequality The classes \(S_{n}^{\ast}(\gamma)\) and \(C_{n}^{\ast}(\gamma)\) are essentially from the classes of starlike and convex functions of complex order, which were considered by Nasr and Aouf [3] and Wiatrowsky [4] respectively (Refer also [5]). Let \(S_{n}(\gamma,\lambda,\beta)\) denote the subclass of \(A(n)\) consisting of functions \(f(z)\) which satisfy the following inequality: \begin{equation} \left \rvert \dfrac{1}{\gamma} \left( \dfrac{\lambda z^{3}f'''(z)+(1+2\lambda)z^{2}f''(z)+zf'(z) }{\lambda z^{2}f''(z)+zf'(z)}-1\right) \right\lvert < \beta, \nonumber \end{equation} where \(z\in U, \gamma\in C\setminus\{0\}, 0\leq\lambda \leq1, 0< \beta \leq1.\) Let \(R_{n}(\gamma,\lambda,\beta)\) denote the subclass of \(A(n)\) consisting of functions \(f(z)\) which satisfy the following inequality \begin{equation} \left \rvert \dfrac{1}{\gamma} \left( \lambda z^{2}f'''(z)+(1+2\lambda)zf''(z)+f'(z)-1\right) \right\lvert < \beta, \nonumber \end{equation} where \(z\in U, \gamma\in C\setminus\{0\}, 0\leq\lambda \leq1, 0< \beta \leq1.\) The class \(S_{n}(\gamma,\lambda,\beta)\) was studied by Kamali and Akbulut [6]. Since \(A\) is the class of functions \(f(z)\) of the form \(f(z)=z+\sum_{k=2}^{\infty}a_{k}z^{k}\) which are analytic in the open unit disk \(U=\{z:\lvert z\rvert< 1\}\). For a function \(f\) in \(A\) we define the following differential operator: \begin{align} D^{0}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)=f(z), \nonumber \end{align} \begin{align} D^{2}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)=D(D^{1}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)), \nonumber \end{align} \begin{align} D^{m}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)=D(D^{m-1}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)). \nonumber \end{align} If \(f\) given by (1), then from (7) we define the following differential operator where \(f(z)\in A, \nu>0,\varrho,\omega,\lambda,\alpha\geq0, m\in N_{0}.\) This operator generalizes certain differential operators such as:

1. \(\nu=1, \varrho=0\) we get $$D^{m}_{\lambda,1,0}(\alpha,\omega)f(z)=z+\sum_{k=2}^{\infty}\left( 1+(k-1)\lambda\omega^{\alpha} \right)^{m} a_{k}z^{k}$$ of Darus and Faisal (2012) (see [7]);
2. \(I\alpha=\omega=\nu=1, \varrho=0\) we get $$D^{m}_{\lambda,1,0}(1,1)f(z)=z+\sum_{k=2}^{\infty}\left( 1+\lambda(k-1) \right)^{m} a_{k}z^{k}$$ of Al-Oboudi (2004) (see [8]);
3. \(\alpha=\omega=\nu=\lambda=1, \varrho=0\) we get $$D^{m}_{1,1,0}(1,1)f(z)=z+\sum_{k=2}^{\infty}\left( k \right)^{m} a_{k}z^{k}$$ of Salagean (1983) (see [9]);
4. \(\alpha=\omega=\nu=1, \lambda=2, \varrho=0\), we get $$D^{m}_{2,1,0}(1,1)f(z)=z+\sum_{k=2}^{\infty}\left( \dfrac{k+1}{2} \right)^{m} a_{k}z^{k}$$ of Uralegaddi and Somanatha (1992) (see [10]).

By using the same processs, we can write the following equalities for the function \(f(z)\) belonging to the class \(A(n)\), \begin{equation} D^{0}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)=f(z) \nonumber \end{equation} \begin{equation} D^{2}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)=D(D^{1}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)) \nonumber \end{equation} \begin{equation} D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)=D(D^{\mho-1}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)). \nonumber \end{equation} If \(f\) given by (2), then from (9) we define the following new differential operator: where \(f\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}.\) Finally, in the term of the generalized S\u{a}l\u{a}gean differential operator, let \(S_{n,\mu}(\gamma,\alpha,\) \(\beta,\lambda,\nu,\varrho,\mho)\) denote the subclass of \(A(n)\) consisting of the functions \( f(z)\) which satisfy the inequality where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha,\mu\geq0, \mho\in N_{0}, z\in U.\) Also, let \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) denote the subclass of \(A(n)\) consisting of the functions \(f(z)\) which satisfy the inequality where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha,\mu\geq0, \mho\in N_{0}, z\in U.\)
Our main work here is to investigate the \((n,\delta)\)-neighborhood of the above said classes i.e. \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) and \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\). Similar work has been seen for different subclasses done by other authors (see for example ([11, 12, 13]) and of course many others.

2. Inclusion relations involving \((n,\delta)\)-neighbourhood

In this section we proved the class relation as well as inclusion relation involving \((n,\delta)\)-neighborhood for the subclasses \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) and \(R_{n,\mu}(\gamma,\) \(\alpha,\beta,\lambda,\nu,\varrho,\mho)\) which depends on the following lemmas.

Lemma 2.1. Let the function \(f(z)\in A(n)\) be defined by (2), then \(f(z)\) is in the class \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) if and only if

where \(f\in A(n),\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U. \)

Proof. Let \(f(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\), then from (11) we have \begin{equation} \left \rvert \dfrac{1}{\gamma} \left( \dfrac{(\mu)z(D^{\mho+3}_{\lambda}(\nu,\alpha,\omega)f(z))^{'} +(1-\mu)z(D^{\mho+2}_{\lambda}(\nu,\alpha,\omega)f(z))^{'}}{(\mu)z(D^{\mho+2}_{\lambda}(\nu,\alpha,\omega)f(z))^{'} +(1-\mu)z(D^{\mho+1}_{\lambda}(\nu,\alpha,\omega)f(z))^{'}}-1\right) \right\lvert < \beta, \nonumber \end{equation} where \(f\in A(n),\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0},\) or \begin{equation} \Re\left( \dfrac{(\mu)z(D^{\mho+3}_{\lambda}(\nu,\alpha,\omega)f(z))^{'} +(1-\mu)z(D^{\mho+2}_{\lambda}(\nu,\alpha,\omega)f(z))^{'}}{(\mu)z(D^{\mho+2}_{\lambda}(\nu,\alpha,\omega)f(z))^{'} +(1-\mu)z(D^{\mho+1}_{\lambda}(\nu,\alpha,\omega)f(z))^{'}}-1\right) >-\beta \lvert\gamma \rvert, \nonumber \end{equation} where \(f\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}.\) This implies that \begin{equation} \Re\left( \dfrac{-\sum_{k=n}^{\infty}{k(\varrho+\lambda)\omega^{\alpha}}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1} a_{k+1}z^{k+1}}{z-\sum_{k=n}^{\infty}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1} a_{k+1}z^{k+1}}\right) >-\beta \lvert\gamma \rvert, \nonumber \end{equation} where \(f\in A(n), \gamma\in C\setminus\{0\} ,\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U,\) after taking the limit when \(z\longrightarrow 1^{-}\) and simplifying, we get \begin{eqnarray*} &&\sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U).\) Conversely, by applying the hypothesis (13) and letting \(\lvert z\rvert=1\) we get \begin{equation} \left \rvert \dfrac{(\mu)z(D^{\mho+3}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))^{'} +(1-\mu)z(D^{\mho+2}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))^{'}}{(\mu)z(D^{\mho+2}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))^{'} +(1-\mu)z(D^{\mho+1}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))^{'}}-1 \right\lvert \nonumber \end{equation} \begin{equation} =\left| \dfrac{-\sum_{k=n}^{\infty}{k(\varrho+\lambda)\omega^{\alpha}}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1} a_{k+1}z^{k+1}}{z-\sum_{k=n}^{\infty}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1} a_{k+1}z^{k+1}}\right| \nonumber \end{equation} \begin{equation} \leq \left|\dfrac{\beta\lvert \gamma\rvert\left[ 1-\sum_{k=n}^{\infty}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1} a_{k+1}\right]}{1-\sum_{k=n}^{\infty}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1} a_{k+1}z^{k+1}}\right|=\beta\lvert \gamma\rvert. \nonumber \end{equation} where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0},z\in U.\) This implies that \(f(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\).

Corollary 2.2. Let the function \(f\) which is defined by (2) be in the class \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\). Then we have \begin{equation} a_{k+1}\leq\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}, k\geq n \nonumber \end{equation} where \(\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}.\)

Lemma 2.3. Let the function \(f(z)\in A(n)\) be defined by (2), then \(f(z)\) is in the class \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) if and only if

where \((f\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\)

Proof. Same as Lemma 2.1.

Theorem 2.4. Let \(f(z)\in A(n)\), then \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\subset N_{n,\delta} (e)\) if

where \(\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\)

Proof. Let \(f(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\), then from (13) we get \begin{eqnarray*} && \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U. \) or \begin{eqnarray*} &&\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\\&&\times\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\sum_{k=n}^{\infty}\rvert a_{k+1}\lvert\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\) This implies that

where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\) By using (13) we have \begin{eqnarray*} && \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+1-1+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0},\) therefore \begin{eqnarray*} &&\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\left(\dfrac{\nu+ n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\sum_{k=n}^{\infty}a_{k+1}\\ &&\leq\beta \lvert\gamma \rvert+(1-\beta \lvert\gamma \rvert)\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\sum_{k=n}^{\infty}a_{k+1}\\ &&\leq\beta \lvert\gamma \rvert+(1-\beta \lvert\gamma \rvert)\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\\ &&\times\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\\ &&\leq\dfrac{\beta \lvert\gamma \rvert \left(\dfrac{\nu+ n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\left(\dfrac{\nu \beta \lvert\gamma \rvert+ n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}, \end{eqnarray*} where \(f\in A(n), \nu\neq0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\) Hence \(\sum_{k=n}^{\infty}(k+1)a_{k+1}\leq\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}=\delta. \) Hence by using (4), we conclude that \( f(z)\in N_{n,\delta}(e) \), this implies that \begin{equation} S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\in N_{n,\delta} (e) \nonumber \end{equation}

Using the same technique of the proof of the Theorem 2.4, we proved the following theorem.

Theorem 2.5. Let \(f(z)\in A(n)\) ,then \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\subset N_{n,\delta} (e)\) if

where \(f\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U. \)

Proof. The proof for this theorem similar to that given above and we omit it.

3. Neighbourhood properties for \({ S^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho))}\) and \({ R^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)}\)

In this section, we determine the neighbourhood for each of the classes \(S^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\) \(\lambda,\nu,\varrho,\mho)\) and \( R^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\). A function \(f(z)\in A(n)\) is said to be in the class \(S^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) if there exists a function \(g(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) such that Similarly, a function \(f(z)\in A(n)\) is said to be in the class \(R^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) if there exists a function \(g(z)\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) satisfying the same inequality \begin{equation} \left| \dfrac{f(z)}{g(z)}-1\right|< 1-\tau, z\in U, \tau\geq0. \nonumber \end{equation}

Theorem 3.1. Let \(g(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) and

where \(f\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U,\) then \begin{equation} N_{n,\delta}(g)\subset S^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho). \nonumber \end{equation}

Proof. Let \(f\in N_{n,\delta}(g)\), then from (3) we can write that\begin{equation} \sum_{k=n}^{\infty}(k+1)\rvert a_{k+1}-b_{k+1}\lvert\leq\delta. \nonumber \end{equation} This implies that \begin{equation} \sum_{k=n}^{\infty}\rvert a_{k+1}-b_{k+1}\lvert\leq\dfrac{\delta}{n+1}. \nonumber \end{equation} Since it is given that \(g(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\), so from (13) we can write that \begin{equation*} \sum_{k=n}^{\infty}b_{k+1}\leq\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}. \nonumber \end{equation*} Now \begin{eqnarray*} &&\left| \dfrac{f(z)}{g(z)}-1\right|< \dfrac{\sum_{k=n}^{\infty}\rvert a_{k+1}-b_{k+1}\lvert}{1-\sum_{k=n}^{\infty}b_{k+1}}\\ &&\leq\dfrac{\delta}{n+1}.\dfrac{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1){\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)-\beta\lvert \gamma\rvert}}\\ &&=\dfrac{\delta\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right){\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)-\beta\lvert \gamma\rvert}}\\ &&=1-\tau, \end{eqnarray*} this implies that \(f\in S^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\), therefore \begin{equation} N_{n,\delta}(g)\subset S^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho). \nonumber \end{equation}

Similarly, by using the same technique of Theorem 3.1 we proved the following theorem.

Theorem 3.2. Let \(g(z)\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho)\) and

where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U,\) then \begin{equation} N_{n,\delta}(g)\subset R^{\tau}_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho). \nonumber \end{equation}

Our next work is to investigate several new results like growth and distortion theorems, Hadamard product, extreme points, integral means inequalities and inclusion properties for the function included in the classes \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \( R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\). Similar work has been seen for different subclasses done by other authors (see for example [14,15, 16, 17, 18, 19, 20, 21]).

4. Growth and distortion theorems

A growth and distortion property of function \(f\) in the respective classes \(S_{n,\mu}(\gamma,\alpha,\) \(\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \( R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) are given as follows:

Theorem 4.1. If the function \(f\) defined by (2) belong to the class \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\) \(\nu,\varrho,\mho,\omega)\) and then for \(\lvert z\rvert< 1\), we have \begin{equation} \lvert f(z)\rvert\leq\lvert z\rvert+\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n+1}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)} \nonumber \end{equation} \begin{equation} \lvert f(z)\rvert\geq\lvert z\rvert-\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n+1}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)} \nonumber \end{equation} where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\) The extremal functions are \begin{equation} f(z)=z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{k+1}, k\geq n.\nonumber \end{equation}

Proof. Let \(f(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\), then from (13) we get \begin{eqnarray*} &&\sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\lvert a_{k+1}\rvert\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U,\) or \begin{eqnarray*} &&\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(f\in A(n), \gamma\in C\setminus\{0\},\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U. \) This implies that \begin{eqnarray*} \sum_{k=n}^{\infty}a_{k+1}\leq\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)} \nonumber \end{eqnarray*} where \(\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}.\) From (13) we have \begin{equation} \lvert f(z)\rvert=\lvert z-\sum_{k=n}^{\infty}a_{k+1}z^{k+1}\rvert, \nonumber \end{equation} or \begin{equation} \lvert f(z)\rvert\geq\lvert z\rvert-\sum_{k=n}^{\infty}\lvert a_{k+1}\rvert \lvert z^{k+1}\rvert. \nonumber \end{equation} This implies that \begin{equation} \lvert f(z)\rvert\geq\lvert z\rvert-\dfrac{\beta\lvert \gamma\rvert }{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z^{n+1}\rvert. \nonumber \end{equation} Similarly \begin{equation} \lvert f(z)\rvert=\lvert z-\sum_{k=n}^{\infty}a_{k+1}z^{k+1}\rvert\leq\lvert z+\sum_{k=n}^{\infty}a_{k+1}z^{k+1}\rvert, \nonumber \end{equation} or \begin{equation} \lvert f(z)\rvert\leq\lvert z\rvert+\sum_{k=n}^{\infty}\lvert a_{k+1}\rvert \lvert z^{k+1}\rvert, \nonumber \end{equation} \begin{equation} \lvert f(z)\rvert\leq\lvert z\rvert+\dfrac{\beta\lvert \gamma\rvert }{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z^{n+1}\rvert. \nonumber \end{equation}

Theorem 4.2. If the function \(f\) defined by (2) belong to the class \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\) \(\nu,\varrho,\mho,\omega)\), then for \(\lvert z\rvert< 1\), we have \begin{equation} \lvert f(z)\rvert\leq\lvert z\rvert+\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n+1}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)} \nonumber \end{equation} \begin{equation} \lvert f(z)\rvert\geq\lvert z\rvert-\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n+1}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)} \nonumber \end{equation} \begin{equation} (f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U). \nonumber \end{equation} The extremal functions are \begin{equation} f(z)=z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)}z^{k+1}, k\geq n. \nonumber \end{equation}

Proof. The proof for this theorem is similar to that given above and we omit it.

Theorem 4.3. If the function \(f\) defined by (2) belong to the class \(S_{n,\mu}(\gamma,\alpha,\beta,\) \(\lambda,\nu,\varrho,\mho,\omega)\), then for \(\lvert z\rvert< 1\) we have \begin{eqnarray*} &&\lvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\rvert\\&&\leq\lvert z\rvert+\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z\rvert^{n+1} \nonumber \end{eqnarray*} \begin{eqnarray*} &&\lvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\rvert\\&&\geq\lvert z\rvert-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z\rvert^{n+1} \nonumber \end{eqnarray*} where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\)

Proof. Let \(f(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\), then from (11) we get \begin{eqnarray*} && \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(f\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U,\) or \begin{eqnarray*} &&\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) \sum_{k=n}^{\infty}\lvert a_{k+1}\rvert\leq\beta \lvert\gamma \rvert, \nonumber \end{eqnarray*} where \(\nu>0,\varrho,\omega,\lambda,\alpha,\mu\geq0, \mho\in N_{0}.\) This implies that \begin{eqnarray*} &&\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) \sum_{k=n}^{\infty}\lvert a_{k+1}\rvert\\&&\leq\sum_{k=n}^{\infty}\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\lvert a_{k+1}\rvert\\&&\leq\beta \lvert\gamma \rvert \nonumber \end{eqnarray*} or \begin{eqnarray*} &&\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\\&&\times\sum_{k=n}^{\infty}\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho }\lvert a_{k+1}\rvert\leq\beta \lvert\gamma \rvert \nonumber \end{eqnarray*} where \(\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0},\) implies that \begin{eqnarray*} &&\sum_{k=n}^{\infty}\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho }\lvert a_{k+1}\rvert\\&&\leq\dfrac{\beta \lvert\gamma \rvert}{\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right).} \nonumber \end{eqnarray*} From (10) we have \begin{equation} \rvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\lvert=\left|z-\sum_{k=n}^{\infty}\left( \dfrac{\nu+(k+1-1)(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) ^{\mho} a_{k+1}z^{k+1}\right|. \nonumber \end{equation} \begin{equation} \rvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\lvert\geq\rvert z\rvert-\sum_{k=n}^{\infty}\left( \dfrac{\nu+(k+1-1)(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho}\lvert a_{k+1}\rvert \lvert z^{k+1}\lvert. \nonumber \end{equation} \begin{eqnarray*} &&\rvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\lvert\\&&\geq\rvert z\rvert-\dfrac{\beta \lvert\gamma \rvert}{\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z^{k+1}\lvert. \nonumber \end{eqnarray*} Similarly we can show that \begin{eqnarray*} &&\rvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\lvert\\&&\leq\rvert z\rvert+\dfrac{\beta \lvert\gamma \rvert}{\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z^{k+1}\lvert. \nonumber \end{eqnarray*}

Theorem 4.4. If the function \(f\) defined by (2) belong to the class \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\) \(\nu,\varrho,\mho,\omega)\), then for \(\lvert z\rvert< 1\), we have \begin{equation} \lvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\rvert\leq\lvert z\rvert+\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)}z\rvert^{n+1}\rvert \nonumber \end{equation} \begin{equation} \lvert D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z)\rvert\geq\lvert z\rvert-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(n+1)}z\rvert^{n+1}\rvert \nonumber \end{equation} where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\)

Theorem 4.5. Let the hypotheses of Theorem 4.1 be satisfied, then \begin{equation} \lvert f'(z)\rvert\leq1+\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)} \nonumber \end{equation} \begin{equation} \lvert f'(z)\rvert\geq1-\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)} \nonumber \end{equation} where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\)

Theorem 4.6. Let the hypotheses of Theorem 4.2 be satisfied, then \begin{equation} \lvert f'(z)\rvert\leq1+\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)} \nonumber \end{equation} \begin{equation} \lvert f'(z)\rvert\geq1-\dfrac{\beta\lvert \gamma\rvert\lvert z\rvert^{n}}{\left(\dfrac{\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)} \nonumber \end{equation} where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\)

Theorem 4.7. Let the hypotheses of Theorem 4.3 be satisfied, then \begin{equation} \lvert (D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))'\rvert\leq1+\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z\rvert^{n} \nonumber \end{equation} \begin{equation} \lvert (D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))'\rvert\geq1-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\left(\dfrac{\beta\lvert\gamma \rvert\nu+n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\lvert z\rvert^{n} \nonumber \end{equation} where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U.\)

Theorem 4.8. Let the hypotheses of Theorem 4.4 be satisfied, then \begin{equation} \lvert (D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))'\rvert\leq1+\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\rvert z\rvert^{n} \nonumber \end{equation} \begin{equation} \lvert (D^{\mho}_{\lambda,\nu,\varrho}(\alpha,\omega)f(z))'\rvert\geq1-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{2\nu+\mu n(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}\rvert z\rvert^{n} \nonumber \end{equation} where \(f(z)\in A(n), \nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}, z\in U. \)

5. Extreme points

In this section we discussed extreme points for functions belonging to the classes \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \( R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\).

Theorem 5.1. (a). Let \(f_{1}(z)=z\) and \begin{equation} f_{i}(z)=z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{i+1}, k\geq n. \nonumber \end{equation} then \(f\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) if and only if it can be expressed in the form \(f(z)=\sum_{i=1}^{\infty}\lambda_{i}f_{i}(z)\) where \(\lambda_{i}\geq0\) and \(\sum_{i=1}^{\infty}\lambda_{i}=1.\)
(b). Let \(f_{1}(z)=z\) and \begin{equation} f_{i}(z)=z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)}z^{i+1}, k\geq n. \nonumber \end{equation} then \(f\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) if and only if it can be expressed in the form \(f(z)=\sum_{i=1}^{\infty}\lambda_{i}f_{i}(z)\) where \(\lambda_{i}\geq0\) and \(\sum_{i=1}^{\infty}\lambda_{i}=1.\)

Proof. Let \(f(z)\in\sum_{i=1}^{\infty}\lambda_{i}f_{i}(z), i=1,2,3,... \lambda_{i}\geq0\) with \(\sum_{i=1}^{\infty}\lambda_{i}=1.\) This implies that \begin{equation} f(z)\in\sum_{i=1}^{\infty}\lambda_{i}f_{i}(z), \nonumber \end{equation} or \begin{eqnarray*} && f(z)=\lambda_{1}(z)+\sum_{i=2}^{\infty}\lambda_{i}\\&&\times\left(z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{i+1}\right), \nonumber \end{eqnarray*} \begin{eqnarray*} &&f(z)=\lambda_{1}(z)+\sum_{i=2}^{\infty}\lambda_{i}(z)-\sum_{i=2}^{\infty}\lambda_{i}\\&&\times\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{i+1} , \nonumber \end{eqnarray*} \begin{eqnarray*} &&f(z)=\sum_{i=1}^{\infty} \lambda_{i}(z)-\sum_{i=2}^{\infty}\lambda_{i}\\&&\times\left( \dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{i+1}\right), \nonumber \end{eqnarray*} \begin{eqnarray*} &&f(z)=(z) -\sum_{i=2}^{\infty}\lambda_{i}\\&&\times\left( \dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu}\right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{i+1}\right). \nonumber \end{eqnarray*}} Since \begin{equation} \sum_{i=2}^{\infty}\left( \dfrac{\lambda_{i}\beta\lvert \gamma\rvert\left[ \left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)\right] }{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{i+1}\right) \nonumber \end{equation} \begin{equation} =\sum_{i=2}^{\infty}\lambda_{i}\beta\lvert \gamma\rvert=\beta\lvert \gamma\rvert\sum_{i=2}^{\infty}\lambda_{i}=\beta\lvert \gamma\rvert(1-\lambda_{1})< \beta\lvert \gamma\rvert. \nonumber \end{equation} The condition (13) for \(f(z)\in\sum_{i=1}^{\infty}\lambda_{i}f_{i}(z)\) is satisfied. Thus \(f\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\) \(\nu,\varrho,\mho,\omega)\). Conversely, we suppose that \(f\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) since \begin{eqnarray*} \lvert a_{k+1}\rvert\leq\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}, k\geq n \nonumber \end{eqnarray*} where \(\nu\neq0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}.\) We put \begin{equation} \lambda_{i}=\dfrac{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\beta\lvert \gamma\rvert}a_{i}, k\geq n \nonumber \end{equation} where \(\nu>0,\varrho,\omega,\lambda,\alpha\geq0, \mho\in N_{0}\) and \( \lambda \nonumber_{1}=1-\sum_{i=2}^{\infty}\lambda_{i},\) then \begin{equation} f(z)=\sum_{i=1}^{\infty}\lambda_{i}f_{i}(z).\nonumber \end{equation}

The proof of the second part of Theorem 5.1 is similar to the first part.

6. Integral means inequalities

For any two functions \(f\) and \(g\) analytic in \(U\), \(f\) is said to be subordinate to \(g\) in \(U\) denoted by \(f\prec g\) if there exists an analytic function \(w\) defined \(U\) satisfying \(w(0)=0\) and \(\lvert w(z)\rvert< 1\) such that \(f(z)=g(w(z)), z\in U\).

In particular, if the function \(g\) is univalent in \(U\), the above subordination is equivalent to \(f(0)=g(0)\) and \(f(U)\subset g(U)\). In 1925, Littlewood [22] proved the following subordination theorem.

Theorem 6.1. If \(f\) and \(g\) are any two functions, analytic in \(U\) with \(f\prec g\) then for \(\mu>0\) and \(z=re^{i\theta} (0< r< 1)\), \begin{equation} \int^{2\pi}_{0}\lvert f(z)\rvert^{\mu}d\theta\leq \int^{2\pi}_{0}\lvert g(z)\rvert^{\mu}d\theta .\nonumber \end{equation}

Theorem 6.2. (a). Let \(f\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \(f_{k}\) be defined by \begin{equation} f_{k}(z)=z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{k+1}, k\geq n. \nonumber \end{equation} if there exists an analytic function \(w(z)\) given by \begin{equation} [w(z)]^{k}=\dfrac{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\beta\lvert \gamma\rvert}\sum_{k=n}^{\infty}a_{k+1}z^{k}, \nonumber \end{equation} then for \(z=re^{i\theta} (0< r< 1)\), \begin{equation} \int^{2\pi}_{0}\lvert f(re^{i\theta})\rvert^{\mu}d\theta\leq \int^{2\pi}_{0}\lvert f_{k}(re^{i\theta})\rvert^{\mu}d\theta \nonumber \end{equation} (b). Let \(f\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \(f_{k}\) be defined by \begin{equation} f_{k}(z)=z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)}z^{k+1}, k\geq n. \nonumber \end{equation} If there exists an analytic function \(w(z)\) given by \begin{equation} [w(z)]^{k}=\dfrac{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)}{\beta\lvert \gamma\rvert}\sum_{k=n}^{\infty}a_{k+1}z^{k}, \nonumber \end{equation} then for \(z=re^{i\theta} (0< r< 1)\), \begin{equation} \int^{2\pi}_{0}\lvert f(re^{i\theta})\rvert^{\mu}d\theta\leq \int^{2\pi}_{0}\lvert f_{k}(re^{i\theta})\rvert^{\mu}d\theta \nonumber \end{equation}

Proof. (a) We have to show that \begin{equation} \int^{2\pi}_{0}\lvert f(re^{i\theta})\rvert^{\mu}d\theta\leq \int^{2\pi}_{0}\lvert f_{k}(re^{i\theta})\rvert^{\mu}d\theta \nonumber \end{equation} or \(\int^{2\pi}_{0}\lvert z-\sum_{k=n}^{\infty} a_{k+1}z^{k+1})\rvert^{\mu}d\theta\) \begin{equation} \leq \int^{2\pi}_{0}\lvert z-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{k+1}\rvert ^{\mu}d\theta \nonumber \end{equation} or \(\int^{2\pi}_{0}\lvert 1-\sum_{k=n}^{\infty} a_{k+1}z^{k})\rvert^{\mu}d\theta\) \begin{equation} \leq \int^{2\pi}_{0}\lvert 1-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{k}\rvert^{\mu}d\theta \nonumber \end{equation} By using Theorem 6.1 it is enough to show that \( 1-\sum_{k=n}^{\infty} a_{k+1}z^{k}\) \begin{equation} < 1-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}z^{k}. \nonumber \end{equation} Now by taking \( 1-\sum_{k=n}^{\infty} a_{k+1}z^{k}\) \begin{equation} = 1-\dfrac{\beta\lvert \gamma\rvert}{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}(w(z))^{k} \nonumber \end{equation} and after simplification we get \begin{equation} [w(z)]^{k}=\dfrac{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\beta\lvert \gamma\rvert}\sum_{k=n}^{\infty}a_{k+1}z^{k}. \nonumber \end{equation} This implies that \(w(0)=0\) and \begin{equation} \lvert[w(z)]^{k}\rvert=\left| \dfrac{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\beta\lvert \gamma\rvert}\sum_{k=n}^{\infty}a_{k+1}z^{k}\right| , \nonumber \end{equation} or \begin{equation} \lvert[w(z)]^{k}\rvert= \dfrac{\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)}{\beta\lvert \gamma\rvert}\sum_{k=n}^{\infty}\lvert a_{k+1}\rvert \lvert z^{k}\rvert. \nonumber \end{equation} By using (13) we get \begin{equation} \lvert[w(z)]^{k}\rvert\leq\lvert z\rvert < 1. \nonumber \end{equation}

The proof of the second part of the Theorem 6.2 is similar to the first part.

7. Inclusion properties

Here we discussed the inclusion properties of the subclasses of analytic functions of complex order denoted by \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \( R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\) \(\omega)\).

Theorem 7.1. (a) Let \(0\leq\alpha_{1}\leq\alpha_{2}\leq1,0\leq\beta_{1}\leq\beta_{2}\leq1,0\leq\lambda_{1}\leq\lambda_{2}\leq1\) and \(0\leq\omega_{1}\leq\omega_{2}\leq1\). Let a function \(f\) be in the class \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) satisfying \begin{equation} \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert \nonumber \end{equation} then
\(\bullet\) \(S_{n,\mu}(\gamma,\alpha_{1},\beta,\lambda,\nu,\varrho,\mho,\omega)\subseteq S_{n,\mu}(\gamma,\alpha_{2},\beta,\lambda,\nu,\varrho,\mho,\omega)\).
\(\bullet\) \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{2},\nu,\varrho,\mho,\omega)\subseteq S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{1},\nu,\varrho,\mho,\omega)\).
\(\bullet\) \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega_{2})\subseteq S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega_{1})\).
\(\bullet\) \(S_{n,\mu_{2}}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\subseteq S_{n,\mu_{1}}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\).
(b) Let \(0\leq\alpha_{1}\leq\alpha_{2}\leq1,0\leq\beta_{1}\leq\beta_{2}\leq1,0\leq\lambda_{1}\leq\lambda_{2}\leq1\) and \(0\leq\omega_{1}\leq\omega_{2}\leq1\). Let a function \(f\) be in the class \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) satisfying \begin{equation} \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +2}\left(\dfrac{2\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1) a_{k+1}\leq\beta \lvert\gamma \rvert \nonumber \end{equation} then
\(\bullet\) \(R_{n,\mu}(\gamma,\alpha_{1},\beta,\lambda,\nu,\varrho,\mho,\omega)\subseteq R_{n,\mu}(\gamma,\alpha_{2},\beta,\lambda,\nu,\varrho,\mho,\omega)\).
\(\bullet\) \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda_{2},\nu,\varrho,\mho,\omega)\subseteq R_{n,\mu}(\gamma,\alpha,\beta,\lambda_{1},\nu,\varrho,\mho,\omega)\).
\(\bullet\) \(R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega_{2})\subseteq R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega_{1})\).
\(\bullet\) \(R_{n,\mu_{2}}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\subseteq R_{n,\mu_{1}}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\).

Proof. (a) To prove \(S_{n,\mu}(\gamma,\alpha_{1},\beta,\lambda,\nu,\varrho,\mho,\omega)\subseteq S_{n,\mu}(\gamma,\alpha_{2},\beta,\lambda,\nu,\varrho,\mho,\omega)\). Since it is given that \(0\leq\alpha_{1}\leq\alpha_{2}\leq1\), implies \begin{equation} \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha_{2}}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha_{2}}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha_{2}}}{\nu} \right) a_{k+1} \nonumber \end{equation} \begin{equation} \leq\sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha_{1}}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha_{1}}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha_{1}}}{\nu} \right) a_{k+1} \nonumber \end{equation} therefore if \(f(z)\in S_{n,\mu}(\gamma,\alpha_{1},\beta,\lambda,\nu,\varrho,\mho,\omega)\) implies \(f(z)\in S_{n,\mu}(\gamma,\alpha_{2},\beta,\lambda,\nu,\varrho,\) \(\mho,\omega)\). This show that \(S_{n,\mu}(\gamma,\alpha_{1},\beta,\lambda,\nu,\varrho,\mho,\omega)\subseteq S_{n,\mu}(\gamma,\alpha_{2},\beta,\lambda,\nu,\varrho,\mho,\omega)\).
Similarly, to prove that \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{2},\nu,\varrho,\mho,\omega)\subseteq S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{1},\nu,\varrho,\mho,\omega)\). Since it is given that \(0\leq\lambda_{1}\leq\lambda_{2}\leq1\) thus \begin{equation} \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda_{1})\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda_{1})\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda_{1})\omega^{\alpha}}{\nu} \right) a_{k+1} \nonumber \end{equation} \begin{equation} \leq\sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda_{2})\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda_{2})\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda_{2})\omega^{\alpha}}{\nu} \right) a_{k+1} \nonumber \end{equation} Therefore, if \(f(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{2},\nu,\varrho,\mho,\omega)\) implies \(f(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{1},\nu,\) \(\varrho,\mho,\omega)\). This show that \(S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{2},\nu,\varrho,\mho,\omega)\subseteq S_{n,\mu}(\gamma,\alpha,\beta,\lambda_{1},\nu,\varrho,\mho,\omega)\).

The proof of the remaining parts of the theorem is similar.

8. Hadamard Product

Theorem 8.1. Let \(f,g\in A(n)\) where \(f(z)\) is given in (2) and \(g(z)=z-\sum_{k=n}^{\infty}b_{k+1}z^{k+1}\), then the modified Hadamard product \(f\ast g\) is defined by \((f\ast g)(z)=z-\sum_{k=n}^{\infty}a_{k+1}b_{k+1}z^{k+1}\). (a) If \(f(z)=z-\sum_{k=n}^{\infty}a_{k+1}z^{k+1}\in S_{n,\mu}(\gamma,\alpha,\beta,\) \(\lambda,\nu,\varrho,\mho,\omega)\) and \(g(z)=z-\sum_{k=n}^{\infty}b_{k+1}z^{k+1}\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) then \((f\ast g)(z)\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\).
(b) If \(f(z)=z-\sum_{k=n}^{\infty}a_{k+1}z^{k+1}\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\) and \(g(z)=z-\sum_{k=n}^{\infty}b_{k+1}z^{k+1}\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\), then \((f\ast g)(z)\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\) \(\mho,\omega)\).

Proof. (a) Since it is given that \(f(z)=z-\sum_{k=n}^{\infty}a_{k+1}z^{k+1}\in S_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\) \(\mho,\omega)\), implies \begin{equation} \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert. \nonumber \end{equation} Similarly \(g(z)=z-\sum_{k=n}^{\infty}b_{k+1}z^{k+1}\in R_{n,\mu}(\gamma,\alpha,\beta,\lambda,\nu,\varrho,\mho,\omega)\), implies \begin{equation} \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) b_{k+1}\leq\beta \lvert\gamma \rvert, \nonumber \end{equation} because \begin{equation} \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) b_{k+1} \nonumber \end{equation} \begin{equation} \leq \sum_{k=n}^{\infty}\left(\dfrac{\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)^{\mho +1}\left(\dfrac{\nu+\mu k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right)(k+1)\left(\dfrac{\beta\lvert\gamma \rvert\nu+k(\varrho+\lambda)\omega^{\alpha}}{\nu} \right) a_{k+1}\leq\beta \lvert\gamma \rvert. \nonumber \end{equation}

Other work regarding differential operators for various problems can be found in [6, 7, 23, 24, 25, 26, 27, 28, 29].

Acknowledgment

The work here is supported by GUP-2017-064.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

The enthusiasm in fluid mechanics is really meaningful in the presence of ``transport phenomena, which is a significant feature in thermal, chemical, and mechanical engineering science. Several physical mechanism exists which can be used to transport thermal energy and chemical species through a phase and across boundaries of the phase. The three mechanisms for the heat transfer are diffusion, convection and radiation. Convection of heat transfer is further classified into three subsequent branches, namely; natural (free), forced, and mixed convection, depending upon the physical system initiating the motion of the fluid.

Free convection flows resulting from the heat and mass transfer driven by the combined buoyancy effect due to temperature and concentration variations have been extensively studied, due to their various applications in geosciences, chemical engineering, and industrial activities as food processing and polymer production. Other multiple areas of applications such as: heat transfer from transmission lines and pipes, heat conduction from electrical devices, heat debauchery from the spiral refrigerator element to surrounding air, heat transfer from a heater, heat transfer in nuclear energy poles, extrusion and wiredrawing, atmospheric and oceanic circulation, etc., have been studied by Jaluria [1]. Free convection problems are usually formulated under different situations like constant surface temperature, ramped temperature at the wall or surface heat flux [2].

Generally, the mass transfer due to the concentration differences affects the rate of heat transfer. The driving force for the free convection is buoyancy so its effects cannot be neglected when the fluid velocity is small and temperature difference between the surface and ambient fluid is large enough [3, 4, 5].

Electrically conducting fluids also have received much attention from the researchers due to their large scale applications in industrial appliances. The magneto hydrodynamics (MHD) has its own practical applications, such as the cooling of nuclear reactors by liquid sodium and induction flow meter, which depends on the potential difference in the fluid in the direction perpendicular to the motion and to the magnetic field [6].

The influence of magnetic field on the free convection flow is significant in liquid metals, electrolytes and ionized gases. The work of Soundalgekar et al. [7] is seemed to be the first on mass transfer and magnetic effects. The study of MHD in the free convection has attracted the interest of many researchers in view of its applications in geophysics and astrophysics [8].

The role of "Fractional Calculus" which is now a day's called calculus of 21th century in modern sciences and engineering is very significant. The tools devised in the subject is used in past few decades to provide solutions to hundreds of real world problems. Researchers are interested to solve scientific problems using the techniques developed in fractional calculus. In the recent times, fractional calculus has been extended to several directions for instance fractional-order multipoles in the electromagnetism, electrochemistry, models in mathematical biology, finance, fluid flows tracers, signal processing in engineering, applied mathematics, bio-engineering and fluid dynamics [9]. Many researchers in fluid dynamics widely used fractional derivative models to study the viscoelastic materials such as polymers in glass transition. It is vital to bring in light the fact that fractional derivative generalizations of one dimensional viscoelastic models have seen to be more useful in modeling the response linear regime [10]. This generalization is in good agreement with the second law of thermodynamics.

The fractional derivative operators used up till now, are Riemann-Liouville fractional derivative and Caputo fractional derivative operator [11]. It is observed by many researchers that application of these operators exhibit difficulties, such as Riemann-Liouville fractional derivative of constant functions is not zero, and its Laplace transform contain terms without physical significance. Caputo fractional derivative has eliminated these difficulties, but the kernel in its definition is a singular function. The results that are been found using these operators are expressed in complicated forms involving some generalized functions.

Recently, Caputo and Fabrizio gave a new expression for fractional derivative operator with an exponential kernel without singularities. The Caputo-Fabrizio temporal-fractional derivative is suitable in the use of the Laplace transform. Shah et al. [12]applied the idea of the Caputo-Fabrizio fractional derivatives to generalize the starting flow of second grade fluid over a vertical plate and obtained the exact solutions using the Laplace transform technique. Some other recent studies can be found in [13, 14, 15, 16, 17] and the references therein.

2. Mathematical formulation of the problem

The unsteady magneto-hydrodynamic flow of viscous incompressible fluid past an infinite vertical plate and constant temperature and variable mass diffusion has been studied. Initially, the plate and the fluid are at the same temperature \(T_{\infty}\) in the stationary condition with concentration level \(C_{\infty}\) at all the points. At time \(t=0^{+}\), the plate is moving with a velocity \(U_{0}f(t)\) in its own plane and the temperature of the plate is constant \(T_{w}\) as well species concentration is raised or lowered to the value \(C_{\infty}+C_{w}g(t)\) with time. \(U_{0}\) is a constants with dimension of velocity while the dimensionless functions \(f(\cdot)\) and \(g(\cdot)\) are piecewise continuous of exponential order at infinity and \(f(0)=g(0)=0\) . We made the following assumptions:

1. All fluid physical properties are considered to be constant except the influence of the body force terms.
2. Applied magnetic field of uniform strength \(B_{0}\) is normal to the plate.
3. The fluids conducting property is supposed to be slight and hence the magnetic Reynolds number is lesser than unity, the induced magnetic field is small in comparison with the transverse magnetic field.
4. It is further supposed that there is no applied voltage, as the electric field is absent.
5. Viscous dissipation and Joule heating in energy equation are neglected.

According to Boussinesq's approximation, the unsteady flow is governed by the following set of equations: with initial and boundary conditions Introducing the following dimensionless variables and parameters \begin{equation*} y^{\ast}=\frac{U_{0}y}{\nu}, u^{\ast}=\frac{u}{U_{0}}, t^{\ast}=\frac{tU_{0}^{2}}{\nu}, T^{\ast}=\frac{T-T_{\infty}}{T_{w}-T_{\infty}}, C^{\ast}=\frac{C-C_{\infty}}{C_{w}-C_{\infty}}, \end{equation*} \begin{equation*} Gr=\frac{g\beta_{r}\nu (T_{w}-T_{\infty})}{U_{0}^{3}},Gm=\frac{g\beta_{c}\nu (C_{w}-C_{\infty})}{U_{0}^{3}}, M=\frac{\sigma B_{0}^{2}\nu}{\rho U_{0}^{2}}, \end{equation*} into Eqs. (1)-(6) and dropping the star notation, we have the following initial-boundary problem with dimensionless initial and boundary conditions Here, we have to developed fractional model, replacing the time derivative in Eqs. (8), (9) and (10), with time-fractional derivatives, we obtain the following fractional differential equations where \(D_{t}^{\alpha}u(y,t)\) represent the Caputo-Fabrizio time-fractional derivative of \(u(y,t),\) defined as [17]

3. Solution of the problem

3.1.Calculation for temperature

Taking Laplace transform of Eqs. (15), \((12)_2\), \((13)_2\) and using initial condition Eq. \((11)_2\) , we obtain where \(\gamma=\frac{1}{1-\alpha}\), \(\overline{T}(y,q)\) , is the Laplace transform of \(T(y,t)\) and \(q\) is the transform variable. The solution of the partial differential equation (18) by using conditions in equation (19), we obtain with inverse Laplace transform where \(F_{1}(y,q,a,b,c)\) and \(f_{1}(y,t,a,b,c)\) are defined in Appendix.

3.2. Calculation for concentration

Taking Laplace transform of Eqs. (16), \((12)_3\), \((13)_3\) and using initial condition Eq. \((11)_3\), we obtain where \(\frac{1}{1-\alpha}\),\(\overline{C}(y,q)\) and \(G(q)\) are the Laplace transform of \(C(y,t)\) respectively \(g(t)\) and \(q\) is the transform variable.
The solution of the partial differential equation (22) by using conditions in equation (23), we obtain Eq. (24) can be written in equivalent form as Inverting Laplace transform we obtain the temperature distribution as where \(g'(t)=L^{-1}\{qG(q)\}\) , \(`` \ast "\) represent the convolution product and \(F_{1}(y,q,a,b,c),\) \(f_{1}(y,t,a,b,c)\) are defined in Appendix.

3.3. Calculation for velocity

Taking Laplace transform of Eqs. (14), \((12)_1\), \((13)_1\), using initial condition Eq. \((11)_1\), by introducing Eqs. (20) and (24), we obtain where \(F(q)\) is the Laplace transform of \(f(t)\) .
The solution of the partial differential equation (27) subject to conditions in equation (28) can be written in the following suitable form as where where \(\displaystyle a_{1}=M-S\), \(\displaystyle a_{2}=a_{3}[(1-Pr)\gamma+a_{1}]\), \(\displaystyle a_{3}=\frac{1}{\alpha\gamma}\), \(\displaystyle a_{4}=\frac{a_{2}}{a_{1}}\), \(a_{5}=\frac{a_{1}-a_{2}a_{3}}{a_{1}a_{3}}\), \(b_{1}=M-\lambda\), \(\displaystyle a_{2}=a_{3}[(1-Sc)\gamma+b_{1}]\), \(\displaystyle b_{3}=\frac{b_{1}}{b_{2}}\), \(\displaystyle b_{4}=\frac{b_{2}-a_{3}b_{1}}{a_{3}b_{2}}\) and \(F_{1}(y,q,a,b,c)\) is defined in the Appendix.
Taking inverse Laplace transform of Eqs. (29), (30) and (31), we obtain where and \(f_{1}(y,t,a,b,c),\) is defined in Appendix and ``\(\ast\)" represent the convolution product.

4. Numerical results and discussions

In order to obtain some information on the fluid flow parameters, we have made several numerical simulations using Mathcad software. Obtained results are presented in the graphs from Figures.1- 6. All the parameters and profiles are dimensionless.
We were interested, to analyze the influence of the fractional parameter \(\alpha\) with different values of time \(t\) , magnetic parameter \(M\) , Prandtl number \(Pr\) and Grashof number \(Gr\) on velocity field in order to study the flow behavior.
In Figure 1, we present the influence of the fractional parameter \(\alpha\) with small time \(t\) on velocity profile. It is observe that by increasing the value of the fractional parameter the velocity is decreases with small values of time and by increasing the time the velocity increase.
In Figure 2, we present the influence of the fractional parameter \(\alpha\) with large time \(t\) on velocity profile. It is observe that by increasing the value of the fractional parameter the velocity is increases with large values of time and by increasing the time the velocity increase.
The effect of magnetic field for small and large values of time on velocity profile is presented in Figure 3, respectively Figure 4. It is observe that by increasing the magnetic field the velocity is decreases as which is expected for small as well as for large time.
The effect of Prandtl number \(Pr\) and Grashof number \(Gr\) are presented in Figs. 5 and 6, respectively. It is observe that the velocity decrease by increasing the Prandtl"number \(Pr\) and increase by increasing the values of \(Gr\).

5. Conclusions

This aim of this article presents, the effects of fractional order derivative and magnetic field on double convection flow of viscous fluid over a moving vertical plate with constant temperature and general concentration. Numerical computations and graphical illustrations are used in order to study the effects of the Caputo-Fabrizio time-fractional parameter \(\alpha\), magnetic parameter \(M\), Prandtl and Grashof number on velocity field. The follow important points are observed

1. By increasing the value of the fractional parameter the velocity is decreases with small values of time and by increasing the time the velocity increase.
2. By increasing the value of the fractional parameter the velocity is increases with large values of time and by increasing the time the velocity increase.
3. By increasing the magnetic field the velocity is decreases.
4. The velocity decrease by increasing the Prandtl number \(Pr\).
5. The velocity increase by increasing the values of \(Gr\).

6. Appendix

$$\displaystyle F_{1}(y,q,a,b,c)=\frac{1}{q}exp\left(-y\sqrt{\frac{cq+b}{aq+1}}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(A1)$$ \(\displaystyle f_{1}(y,t;a,b,c)=L^{-1}\{F_{1}(y,q,a,b,c)\}=\) \begin{equation*} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\begin{cases} \displaystyle \frac{c}{a}e^{-\frac{t}{a}}\int_{0}^{t}erfc\left(\frac{y}{2\sqrt{u}}\right)e^{-\frac{cu}{a}}I_{0}\left(\frac{2}{a}\sqrt{(c-ab)ut}\right)du \\ \displaystyle +\frac{b}{a}\int_{0}^{\infty} \int_{0}^{t}erfc\left(\frac{y}{2\sqrt{u}}\right)e^{-\frac{cu+\tau}{a}}I_{0}\left(\frac{2}{a}\sqrt{(c-ab)u\tau}\right)d\tau du ,\,\,\,0< \alpha< 1,\,\,\,\,\,(A2)\\ \displaystyle \psi(y,t,a,b),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha=1.\\ \end{cases} \end{equation*}

Competing Interests

The author(s) do not have any competing interests in the manuscript.