1. Introduction

The term "convexity" is the most important, interesting, natural, and fundamental notations in mathematics and was used for the first time widely in the classical book by Hardy, Little, and Polya (see [1]). In recent times, the theory of convexity has played a very fascinating and amazing role in the world of science, of course, no one can refuse its significance and importance. Many researchers always try to do and use new ideas for the enjoyment and beautification of convex analysis. This theory provides us with interesting and powerful numerical tools, on the basis of these tools, we solve a lot of problems that appear in mathematics. During the last century, many researchers have contributed to the theory of convexity. The theory of convexity and its generalizations also play a magnificent role in the analysis of extremum problems. For the applications and interesting literature about convex analysis, readers refer to [2].

The theory of convexity also played an important and central tool in the development of the theory of inequalities. The subject of "Inequalities" is a very attractive and captivating field of research. The theory of inequalities is a subject of many mathematician's work in the last century. Many mathematicians always keep and continually try to do their work in the field of inequalities with hardworking and to collaborate different ideas and concepts. Thus the theory of inequalities and algorithms may be regarded as an independent area of mathematics. In recent years, due to its diverse and widespread applications, the ideas about the topics of convexity and integral inequalities have been extended, improved, and generalized in many different ways and the researchers are always inspired by the relationship of these two fields and consequently, many new inequalities have been obtained via the convexity property of the functions. For the importance of inequalities, interested readers are refer to [3].

The aim of this paper is to introduce a new family of convex functions namely \(n\)-polynomial generalized convex functions of Raina type. Interesting algebraic properties, several new integral inequalities, example with logic, and applications to means via the newly introduced class of functions are provided.

2. Preliminaries

In this section we recall some known concepts.

Definition 1.[4] A function \(\Psi: I\rightarrow\mathbb{R}\) is said to be convex, if

holds \(\forall \varrho_{1},\varrho_{2}\in I\) and \(\kappa\in[0,1].\)

The most important inequality concerning convex function is Hermite-Hadamard inequality given as:

Theorem 1. If \(\Psi:[\varrho_{1},\varrho_{2}]\rightarrow \mathbb{R}\) is a convex function, then

The double inequality (2) is in reverse order if \(\Psi\) is a concave function. The researchers have shown keep interest in above inequality and as a result various generalizations and improvements have been appeared in the literature. Due to widespread views and applications, this inequality has a lot of importance in the field of analysis.

In 2005, Raina [5] introduced a class of functions defined formally by

where \(\sigma=(\sigma(0),\ldots,\sigma(k),\ldots)\) and \(\rho, \lambda>0, |z|< R\). The above class of function is the generalization of classical Mittag-Leffler function and the Kummer function, if \(\rho=1, \lambda=0\) and \(\sigma(k)=\frac{(\alpha)_{k}(\beta)_{k}}{(\gamma)_{k}}\) for \(k=0,1,2,\ldots.\)

The above parameters \(\alpha, \beta\) and \(\gamma\) can take arbitrary real or complex values (provided that \(\gamma\neq0,-1,-2,\ldots),\) and the symbol \(\alpha_{k}\) represents the quantity

\begin{equation*} (\alpha)_{k}=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}=\alpha(\alpha+1)\ldots(\alpha+k-1),\;\;\;k=0,1,2,\ldots, \end{equation*} and restrict its domain to \(|z|\leq1\) (with \(z\in \mathbb{C}\)), then we have the classical hypergeometric function, that is \begin{equation*} \;{\mathcal{F}}(\alpha, \beta; \gamma;z)=\sum_{k=0}^{+\infty}\frac{(\alpha)_{k}(\beta)_{k}}{k!(\gamma)_{k}}z^{k}. \end{equation*} Also, if \(\sigma=(1,1,\ldots)\) with \(\rho=\alpha,\, (Re(\alpha)>0), \lambda=1\), then \begin{equation*} E_{\alpha}(z)=\sum_{k=0}^{+\infty}\frac{z^{k}}{\Gamma(1+\alpha k)}. \end{equation*} The above function is called a classical Mittag-Leffler function.

Cortez established the new class of set and function involving Raina's function in [6,7], which is said to be generalized convex set and convex function.

Definition 2. [8] Let \(\sigma=(\sigma(0),\ldots,\sigma(k),\ldots)\) and \(\rho,\lambda> 0\). A set \(X\neq\emptyset\) is said to be generalized convex, if

for all \(\varrho_{1},\varrho_{2}\in X\) and \({\kappa}\in[0,1].\)

Definition 3. [8] Let \(\sigma\) denote a bounded sequence then \(\sigma=(\sigma(0),\ldots,\sigma(k),\ldots)\) and \(\rho,\lambda>0\). If \(\Psi:X\rightarrow \mathbb{R}\) satisfies the following inequality

for all \(\varrho_{1},\varrho_{2}\in X,\) where \(\varrho_{1}< \varrho_{2}\) and \({\kappa}\in[0,1],\) then \(\Psi\) is called generalized convex function.

Remark 1. We have \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}>0,\) and so we obtain Definition 1.

Condition 1. Let \(X\subseteq \mathbb{R}\) be an open generalized convex subset with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\cdot)\). For any \(\varrho_{1},\varrho_{2}\in X\) and \({\kappa}\in[0,1],\) \begin{equation*} \;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\Big(\varrho_{2}-(\varrho_{2}+{\kappa}\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\Big)=-{\kappa}\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}), \end{equation*} \begin{equation*} \;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\Big(\varrho_{1}-\left(\varrho_{2}+{\kappa}\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\right)\Big)=(1-{\kappa})\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}). \end{equation*}

Note that, for every \(\varrho_{1},\,\varrho_{2}\in X\) and for all \({\kappa}_{1},\,{\kappa}_{2}\in [0,1]\) from Condition 1, we have

Definition 4. [9] Let \(\forall \varrho_{1},\varrho_{2}\in I,\) \(n\in\mathbb{N}\) and \(\kappa\in[0,1]\), then an inequality of the form

is called an \(n\)-polynomial convex function.

Definition 5. [10] Two functions \(\Psi\) and \(\Phi\) are said to be similarly ordered, if

Theorem 2. [11] Let \(p>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). If \(\Psi,\Phi \in \mathbb{R}\) defined on \([\varrho_{1},\varrho_{2}]\) and if \(\vert \Psi\vert^{p}, \vert \Phi \vert^{q}\) are \(L[\varrho_{1},\varrho_{2}]\), then \begin{align*} \int_{\varrho_{1}}^{\varrho_{2}}\vert \Psi(x)\Phi(x) \vert dx \leq& \frac{1}{\varrho_{2}-\varrho_{1}}\left\{\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(\varrho_{2}-x)\vert \Psi(x)\vert^{p}dx\bigg)^{\frac{1}{p}}\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(\varrho_{2}-x)\vert \Phi(x)\vert^{q}dx\bigg)^{\frac{1}{q}}\right. \\&\left. +\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(x-\varrho_{1})\vert \Psi(x)\vert^{p}dx\bigg)^{\frac{1}{p}}\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(x-\varrho_{1})\vert \Phi(x)\vert^{q}dx\bigg)^{\frac{1}{q}}\right\}. \end{align*}

Theorem 3. [12] Let \(q\geq1\). If \(\Psi,\Phi\in \mathbb{R}\) defined on \([\varrho_{1},\varrho_{2}]\) and if \(\vert \Psi \vert, \vert \Psi \vert\vert \Phi \vert^{q}\) are \(L[\varrho_{1},\varrho_{2}]\), then \begin{align*} \int_{\varrho_{1}}^{b}\vert \Psi(x)\Phi(x) \vert dx \leq& \frac{1}{\varrho_{2}-\varrho_{1}}\left\{\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(\varrho_{2}-x)\vert \Psi(x)\vert dx\bigg)^{1-\frac{1}{q}}\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(\varrho_{2}-x)\vert \Psi(x) \vert\vert \Phi(x)\vert^{q}dx\bigg)^{\frac{1}{q}}\right. \\& \left. +\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(x-\varrho_{1})\vert \Psi(x)\vert dx\bigg)^{1-\frac{1}{q}}\bigg(\int_{\varrho_{1}}^{\varrho_{2}}(x-\varrho_{1})\vert \Psi(x)\vert\vert \Phi(x)\vert^{q}dx\bigg)^{\frac{1}{q}}\right\}. \end{align*}

Owing to the aforementioned trend and energized by the in-progress research activities in this captivating field, we organize the paper in the following pattern. In Section 3, we give the idea and algebraic properties of \(n\)-polynomial generalized convex functions of Raina type. In Section 4 and 5, using the newly introduced idea, we attain the new version of Hermite-Hadamard inequality and some of its refinements. In Section 6, we will add some applications and conclusion.

3. \(n\)-polynomial generalized convex functions of Raina type and its Properties

In this section, we introduce a new family of convex functions namely \(n\)-polynomial generalized convex functions of Raina type and also will study some of its properties. One important thing to keep in mind throughout the paper, \(n\)-poly represents \(n\)-polynomial.

Definition 6. Let \(\mathbb{X}\in \mathbb{R}\) be a nonempty generalized convex set with respect to \({\mathcal{F}}_{\rho,\lambda}^{\sigma}:\mathbb{X}\times\mathbb{X}\rightarrow \mathbb{R}\). Then a nonnegative function \(\Psi:\mathbb{X} \rightarrow \mathbb{R}\) is called \(n\)-poly generalized convex function of Raina type, if

holds for every \(\varrho_{1},\varrho_{2}\in \mathbb{X},\) \(n\in\mathbb{N}\), \(\kappa\in[0,1]\) , \(\sigma=(\sigma(0),\ldots,\sigma(k),\ldots)\), \(\rho, \lambda>0\) and \(|\varrho_{1}-\varrho_{2}|< R\).

Remark 2. It is easy to show that, if nonnegative function \(\Psi\) is generalized convex function of Raina type then nonnegative function \(\Psi\) is \(n\)-poly generalized convex function of Raina type. Indeed for all \(\kappa\in[0,1],\) then we attain the following inequalities \begin{equation*} \kappa\leq \frac{1}{n}\sum_{s=1}^{n}[1-(1-\kappa)^{s}]\;\;\;\mbox{and}\;\;\;{1-\kappa}\leq\frac{1}{n}\sum_{s=1}^{n}[1-\kappa^{s}]. \end{equation*} This mean that, the new family of \(n\)-poly generalized convex functions of Raina type is very larger with respect the known class of functions, like \(n\)-poly convex functions. This is an advantage of the proposed new Definition 6.

Example 1. Since \(\Psi(x)=e^{x}\) is nonnegative convex function, so obviously it is a generalized convex function of Raina type. So according to Remark 2, it is \(n\)-poly generalized convex function of Raina type.

Theorem 4. Let \( \Psi, \Phi:[\varrho_{1}, \varrho_{2}]\rightarrow \mathbb{R} .\) If \( \Psi \) and \( \Phi \) are two \(n\)-poly generalized convex functions of Raina type, then

• (i) \( \Psi + \Phi \) is an \(n\)-poly generalized convex function of Raina type.
• (ii) For nonnegative real number \(c,\) \(c\Psi\) is an \(n\)-poly generalized convex function of Raina type.

Proof.

• (i) Since \( \Psi \) and \(\Phi\) be two \(n\)-poly generalized convex functions of Raina type, then \begin{align*} \left(\Psi+\Phi\right)(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) & =\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))+\Phi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) \\& \leq \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Psi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Psi\left( \varrho_{2}\right)\\ &\;\;\;+\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Phi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Phi\left( \varrho_{2}\right). \\& =\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \left[\Psi\left( \varrho_{1}\right)+\Phi\left( \varrho_{1}\right)\right]+\frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \left[\Psi\left( \varrho_{2}\right)+\Phi\left( \varrho_{2}\right)\right] \\& =\ \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] (\Psi+\Phi)(\varrho_{1})+\frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] (\Psi+\Phi)(\varrho_{2}). \end{align*}
• (ii) Since \( \Psi \) be an \(n\)-poly generalized convex function of Raina type, then \begin{align*} \left(c\Psi\right)(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) & \leq c\bigg[\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Psi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Psi\left( \varrho_{2}\right)\bigg] \\& =\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right]c\Psi\left( \varrho_{1}\right)+ \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] c\Psi\left( \varrho_{2}\right) \\& =\ \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \left(c\Psi\right)\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \left(c\Psi\right)\left( \varrho_{2}\right), \end{align*} which completes the proof.

Theorem 5. Suppose \(\Psi:I\rightarrow J\) be \(n\)-poly generalized convex function of Raina type and \(\Phi:J\rightarrow \mathbb{R}\) is non-decreasing function. Then the composition of two functions i.e., \(\Phi\circ \Psi:I\rightarrow \mathbb{R}\) is an \(n\)-poly generalized convex function of Raina type.

Proof. Let \(\kappa \in [0,1]\) and for all \(\varrho_{1},\varrho_{2} \in I,\) we have \begin{align*} \left(\Phi\circ \Psi\right)(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) & =\Phi(\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))) \\& \leq \Phi\bigg[\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Psi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Psi\left( \varrho_{2}\right)\bigg] \\& \leq\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Phi(\Psi\left( \varrho_{1}\right)) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Phi(\Psi\left( \varrho_{2}\right)) \\& =\ \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] (\Phi\circ \Psi)\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] (\Phi\circ \Psi)\left( \varrho_{2}\right). \end{align*} This completes the proof.

Remark 3.

• (i) Consider \(n=1\) in above Theorem 5, then we attain the following inequality \begin{equation*} \left(\Phi\circ \Psi\right)(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) \leq \kappa (\Phi\circ \Psi)\left( \varrho_{1}\right) + (1-{\kappa}) (\Phi\circ \Psi)\left( \varrho_{2}\right). \end{equation*}
• (ii) If we put \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in above Theorem 5, then we attain the following inequality \begin{equation*} \left(\Phi\circ \Psi\right)(\kappa\varrho_{1}+(1-\kappa)\varrho_{2}) \leq \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] (\Phi\circ \Psi)\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] (\Phi\circ \Psi)\left( \varrho_{2}\right). \end{equation*}
• (iii) If we put \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in above Theorem 5, then we attain the following inequality \begin{equation*} \left(\Phi\circ \Psi\right)(\kappa\varrho_{1}+(1-\kappa)\varrho_{2}) \leq \kappa (\Phi\circ \Psi)\left( \varrho_{1}\right) + (1-{\kappa}) (\Phi\circ \Psi)\left( \varrho_{2}\right). \end{equation*}

Theorem 6. Let \(0< \varrho_{1}< \varrho_{2},\) \(\Psi_{j}: [\varrho_{1},\varrho_{2}]\rightarrow[0,+\infty)\) be a family of \(n\)-poly generalized convex functions of Raina type and \(\Psi(u)=\sup_{j}\Psi_{j}(u)\). Then \(\Psi\) is an \(n\)-poly generalized convex function of Raina type and \(U=\{u\in[\varrho_{1} ,\varrho_{2}]:\Psi(u)< +\infty\}\) is an interval.

Proof. Let \(\varrho_{1},\varrho_{2}\in U\) and \(\kappa\in[0,1],\) then \begin{align*} \Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) & =\sup_{j}\Psi_{j}(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) \\& \leq \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \sup_{j}\Psi_{j}\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \sup_{j}\Psi_{j}\left( \varrho_{2}\right) \\& =\ \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Psi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Psi\left( \varrho_{2}\right)< +\infty. \end{align*} This completes the proof.

Remark 4. Taking \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in above Theorem 6, then we attain Theorem 3 in [9].

Theorem 7. Suppose \( \Psi, \Phi:[\varrho_{1}, \varrho_{2}]\rightarrow \mathbb{R} .\) If \( \Psi \) and \( \Phi \) are two \(n\)-poly generalized convex functions of Raina type. If the above defined functions \(\Psi\) and \(\Phi\) are similarly ordered and \(\frac{1}{n}\sum\limits_{s=1}^{n}\left[1-(1-\kappa)^{s}\right]+\frac{1}{n}\sum\limits_{s=1}^{n}\left[1-{\kappa}^{s}\right]\leq1,\) then the product \(\Psi\Phi\) is also an \(n\)-poly generalized convex function of Raina type.

Proof. Since \(\Psi\) and \(\Phi\) are two \(n\)-poly generalized convex functions of Raina type, then \begin{align*} \Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))&\Phi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) \\ \leq& \bigg[\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Psi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Psi\left( \varrho_{2}\right)\bigg] \\& \times\bigg[\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Phi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Phi\left( \varrho_{2}\right).\bigg] \\ \leq& \frac{1}{n^{2}}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right]^{2} \Psi(\varrho_{1})\Phi(\varrho_{1})+\frac{1}{n^{2}}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right]^{2}\Psi(\varrho_{2})\Phi(\varrho_{2}) \\& +\frac{1}{n^{2}}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right]\left[1-{\kappa}^{s}\right] [\Psi(\varrho_{1})\Phi(\varrho_{2})+\Psi(\varrho_{2})\Phi(\varrho_{1})] \\ \leq& \frac{1}{n^{2}}\sum_{s=1}^{n}[1-(1-\kappa)^{s}]^{2}\Psi(\varrho_{1})\Phi(\varrho_{1})+\frac{1}{n^{2}}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right]^{2} \Psi(\varrho_{2})\Phi(\varrho_{2}) \\& +\frac{1}{n^{2}}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right]\left[1-{\kappa}^{s}\right][\Psi(\varrho_{1})\Phi(\varrho_{1})+\Psi(\varrho_{2})\Phi(\varrho_{2})] \\ =&\bigg[\frac{1}{n}\sum_{s=1}^{n}[1-(1-\kappa)^{s}]\Psi(\varrho_{1})\Phi(\varrho_{1})+\frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right]\Psi(\varrho_{2})\Phi(\varrho_{2})\bigg] \\&\times\bigg(\frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right]+ \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right]\bigg) \\ \leq& \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right]\Psi(\varrho_{1})\Phi(\varrho_{1})+\frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right]\Psi(\varrho_{2})\Phi(\varrho_{2}). \end{align*} This shows that the product of two \(n\)-poly generalized convex functions of Raina type is again an \(n\)-poly generalized convex function of Raina type.

Remark 5.

• (i) Taking \(n=1\) in above Theorem 7, then we obtain \begin{equation*} \Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\Phi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})) \leq \kappa \Psi\left( \varrho_{1}\right)\Phi(\varrho_{1}) + (1-\kappa) \Psi\left( \varrho_{2}\right)\Phi(\varrho_{2}). \end{equation*}
• (ii) Taking \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in above Theorem 7, then we obtain \begin{equation*} \Psi(\kappa\varrho_{1}+(1-\kappa)\varrho_{2})\Phi(\kappa\varrho_{1}+(1-\kappa)\varrho_{2}) \leq \frac{1}{n}\sum_{s=1}^{n}\left[1-(1-\kappa)^{s}\right] \Psi\left( \varrho_{1}\right) + \frac{1}{n}\sum_{s=1}^{n}\left[1-{\kappa}^{s}\right] \Psi\left( \varrho_{2}\right). \end{equation*}
• (iii) Taking \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in above Theorem 7, then we obtain \begin{equation*} \Psi(\kappa\varrho_{1}+(1-\kappa)\varrho_{2})\Phi(\kappa\varrho_{1}+(1-\kappa)\varrho_{2}) \leq \kappa \Psi\left( \varrho_{1}\right)\Phi(\varrho_{1}) + (1-\kappa) \Psi\left( \varrho_{2}\right)\Phi(\varrho_{2}). \end{equation*}

4. Hermite-Hadamard type inequality for \(n\)-polynomial generalized convex functions of Raina type

The focus of this section is to establish Hermite-Hadamard inequality for \(n\)-poly generalized convex functions of Raina type.

Theorem 8. Let \(\Psi:[\varrho_{1},\varrho_{2}]\in\mathbb{R} \) be an \(n\)-poly generalized convex function of Raina type, if \(\varrho_{1}< \varrho_{2} \) and \(\Psi\in L[\varrho_{1},\varrho_{2}]\), then \begin{align*} &\frac{1}{2}\bigg(\frac{n}{n+2^{-n}-1}\bigg)\Psi(\varrho_{2}+\frac{1}{2}\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\leq\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx \leq\frac{\Psi(\varrho_{1})+\Psi(\varrho_{2})}{n}\sum_{s=1}^{n}\frac{s}{s+1}. \end{align*}

Proof. From the property of an \(n\)-poly generalized convex function of Raina type \(\Psi\), we have that \begin{equation*} \Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\leq\frac{1}{n}\sum_{s=1}^{n}[1-(1-\kappa)^{s}]\Psi(\varrho_{1})+\frac{1}{n}\sum_{s=1}^{n}[1-\kappa^{s}]\Psi(\varrho_{2}) \end{equation*} \begin{equation*} \int_{0}^{1}\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa\leq\frac{\Psi(\varrho_{1})}{n}\sum_{s=1}^{n}\int_{0}^{1}[1-(1-\kappa)^{s}]d\kappa+\frac{\Psi(\varrho_{2})}{n}\sum_{s=1}^{n}\int_{0}^{1}[1-\kappa^{s}]d\kappa \end{equation*} but \begin{equation*} \int_{0}^{1}\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa=\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx \end{equation*} so \begin{equation*} \frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\leq\frac{\Psi(\varrho_{1})+\Psi(\varrho_{2})}{n}\sum_{s=1}^{n}\frac{s}{s+1}. \end{equation*} This completes the proof of right side of above inequality. For the left side using the property of an \(n\)-poly generalized convex function of Raina type and Condition A for \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\) and integrating over \([0,1]\), we have \begin{align*} \Psi\Big(\varrho_{2}+&\frac{1}{2}\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\Big) \\& =\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})+\frac{1}{2}\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{2}+(1-\kappa)\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})- (\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))) \\& \leq \frac{1}{n}\sum_{s=1}^{n}\bigg[1-\bigg(\frac{1}{2}\bigg)^{s}\bigg]\bigg[\int_{0}^{1}\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa+\int_{0}^{1}\Psi(\varrho_{2}+(1-\kappa)\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa\bigg] \\& \leq\frac{1}{n}\sum_{s=1}^{n}\bigg[1-\bigg(\frac{1}{2}\bigg)^{s}\bigg] \frac{2}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx \\& \leq\bigg[\frac{n+2^{-n}-1}{n}\bigg]\frac{2}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx. \end{align*} This completes the proof.

Corollary 1. If we put \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 8, then we get Hermite-Hadamard inequality in [13].

Corollary 2. If we put \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 8, then we get inequality (3.1) in [9].

Remark 6. Under the assumption of Theorem 8, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} &\frac{1}{2}\bigg(\frac{n}{n+2^{-n}-1}\bigg)\Psi(\varrho_{2}+\frac{1}{2}E_{\alpha}(\varrho_{1}-\varrho_{2}))\leq\frac{1}{E_{\alpha}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2})}\Psi(x)dx \leq\frac{\Psi(\varrho_{1})+\Psi(\varrho_{2})}{n}\sum_{s=1}^{n}\frac{s}{s+1}. \end{align*}

5. New inequalities for \(n\)-polynomial generalized convex functions of Raina type

The intention of this section is to derived the refinements of Harmite-Hadamard type inequalities for \(n\)-poly generalized convex functions of Raina type.

Lemma 1. Let \(X\subseteq \mathbb{R}\) be an generalized convex subset with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}: X \times X \rightarrow\mathbb{R}\) and \(\varrho_{1}, \varrho_{2} \in X\) with \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\neq 0\). Suppose that \(\Psi: X \rightarrow\mathbb{R}\) is a differentiable function. If \(\Psi\) is integrable on the \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\), then the following equality holds \begin{align*} -&\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}+\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx \\& \;\;\;\;\;\; =\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\int_{0}^{1}(1-2\kappa)\Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa. \end{align*}

Proof. Suppose that \(\varrho_{1}, \varrho_{2} \in X\). Since \(X\) is generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\) , for every \(\kappa \in[0,1]\), we have \(\varrho_{2}+\kappa \;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\in X\). Integrating by parts implies that \begin{align*} \int_{0}^{1}(1-2\kappa)\Psi'&(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa \\& =\bigg[\frac{(1-2\kappa)\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\bigg]_{0}^{1} +\frac{2}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}- \varrho_{2})} \int_{0}^{1}\Psi(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa \\& =-\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}+\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx. \end{align*} This completes the proof.

Theorem 9. Suppose \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\) and \(\Psi:I^{\circ}\subseteq\mathbb{R}\rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\), \( \varrho_{1},\varrho_{2} \in I^{\circ}\) with \(\varrho_{1}< \varrho_{2}\) and \(\Psi'\) is integrable function on the interval \([\varrho_{1},\varrho_{2}]\). Suppose \(\vert \Psi' \vert\) is an \(n\)-poly generalized convex function of Raina type on \(L[\varrho_{1},\varrho_{2}]\), then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{n}\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg] A\bigg(\vert \Psi'(\varrho_{1})\vert, \vert \Psi'(\varrho_{2})\vert \bigg), \end{align*} holds for \(\kappa\in[0,1]\), where A(.,.) is Arithmetic mean.

Proof. Suppose that \( \varrho_{1},\varrho_{2}\in I^{\circ} \). Since \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\), for any \(\kappa\in[0,1],\) we have \( \varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\in I^{\circ} \).

From Lemma 1, \(n\)-poly generalized convex function of Raina type of \(\vert \Psi' \vert\) and properties of modulus, we have

\begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\bigg\vert\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\int_{0}^{1}(1-2\kappa)\Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\int_{0}^{1}\vert 1-2\kappa\vert \bigg(\frac{1}{n}\sum_{s=1}^{n}[1-(1-\kappa)^{s}]\vert \Psi'(\varrho_{1})\vert+\frac{1}{n}\sum_{s=1}^{n}[1-\kappa^{s}]\vert \Psi'(\varrho_{2})\vert\bigg)d\kappa \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2n}\bigg(\vert \Psi'(\varrho_{1})\vert\int_{0}^{1}\vert 1-2\kappa \vert\sum_{s=1}^{n}[1-(1-\kappa)^{s}]d\kappa+\vert \Psi'(\varrho_{2})\vert\int_{0}^{1}\vert 1-2\kappa \vert\sum_{s=1}^{n}\bigg([1-\kappa^{s}]d\kappa\bigg) \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2n}\bigg(\vert \Psi'(\varrho_{1})\vert\sum_{s=1}^{n}\int_{0}^{1}\vert 1-2\kappa \vert[1-(1-\kappa)^{s}]d\kappa+\vert \Psi'(\varrho_{2})\vert\sum_{s=1}^{n}\int_{0}^{1}\vert 1-2\kappa \vert[1-\kappa^{s}]d\kappa\bigg) \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2n}\bigg(\vert \Psi'(\varrho_{1})\vert\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg]+\vert \Psi'(\varrho_{2})\vert\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg]\bigg) \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{n}\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg]A\bigg(\vert \Psi'(\varrho_{1})\vert,\vert \Psi'(\varrho_{2})\vert\bigg). \end{align*} This completes the proof.

Corollary 3. If \(n=1\) in above Theorem 9, then \begin{equation} \bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{4}A\bigg(\vert \Psi'(\varrho_{1})\vert,\vert \Psi'(\varrho_{2})\vert\bigg). \end{equation}

Corollary 4. If we put \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 9, we get inequality (4.1) in [9].

Corollary 5. If we put \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 9, then we get Corollary 1 in [9].

Remark 7. Under the assumption of Theorem 9, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{E_{\alpha}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\;\;\; \leq\frac{E_{\alpha}(\varrho_{1}-\varrho_{2})}{n}\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg] A\bigg(\vert \Psi'(\varrho_{1})\vert, \vert \Psi'(\varrho_{2})\vert \bigg). \end{align*}

Theorem 10. Suppose \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\) and \(\Psi:I^{\circ}\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\), \( \varrho_{1},\varrho_{2} \in I^{\circ}\) with \(\varrho_{1}< \varrho_{2}, q>1, \frac{1}{p}+\frac{1}{q}=1\) and \(\Psi'\) is integrable function on the interval \([\varrho_{1},\varrho_{2}]\). Suppose \(\vert \Psi' \vert^{q}\) is an \(n\)-poly generalized convex function of Raina type on \(L[\varrho_{1},\varrho_{2}]\), then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\;\;\; \leq \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{p+1}\bigg)^{\frac{1}{p}}\bigg(\frac{2}{n}\sum_{s=1}^{n}\frac{s}{s+1}\bigg)^{\frac{1}{q}}A^{\frac{1}{q}} \bigg(\vert \Psi'(\varrho_{1})\vert^{q},\vert \Psi'(\varrho_{2})\vert^{q}\bigg), \end{align*} holds for \(\kappa\in[0,1]\), where A(x,y) is Arithmetic mean.

Proof. Suppose that \( \varrho_{1},\varrho_{2}\in I^{\circ} \). Since \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\), for any \(\kappa\in[0,1],\) we have \( \varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\in I^{\circ} \).

From Lemma 1, Hölder's integral inequality, \(n\)-poly generalized convex function of Raina type of \(\vert \Psi' \vert^{q}\) and properties of modulus, we have

\begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\& \;\;\;\;\;\; \leq\bigg\vert\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\int_{0}^{1}(1-2\kappa)\Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\int_{0}^{1}\vert1-2\kappa \vert^{p}\bigg)^{\frac{1}{p}}\bigg(\int_{0}^{1}\vert \Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert^{q}d\kappa\bigg)^{\frac{1}{q}} \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{p+1}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\int_{0}^{1}\sum_{s=1}^{n}[1-(1-\kappa)^{s}]d\kappa+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\int_{0}^{1}\sum_{s=1}^{n}[1-\kappa^{s}]d\kappa\bigg)^{\frac{1}{q}} \\&\;\;\;\;\;\; \leq \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{p+1}\bigg)^{\frac{1}{p}}\bigg(\frac{2}{n}\sum_{s=1}^{n}\frac{s}{s+1}\bigg)^{\frac{1}{q}}A^{\frac{1}{q}} \bigg(\vert \Psi'(\varrho_{1})\vert^{q},\vert \Psi'(\varrho_{2})\vert^{q}\bigg). \end{align*} This completes the proof.

Corollary 6. If we put \(n=1\) in Theorem 10, then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{p+1}\bigg)^{\frac{1}{q}}A^{\frac{1}{q}}\bigg(\vert \Psi'(\varrho_{1})\vert,\vert \Psi'(\varrho_{2})\vert\bigg). \end{align*}

Corollary 7. If we put \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 10, then we get inequality (4.2) in [9].

Corollary 8. If we put \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 10, then we get Corollary 2 in [9].

Remark 8. Under the assumption of Theorem 10, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{E_{\alpha}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq \frac{E_{\alpha}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{p+1}\bigg)^{\frac{1}{p}}\bigg(\frac{2}{n}\sum_{s=1}^{n}\frac{s}{s+1}\bigg)^{\frac{1}{q}}A^{\frac{1}{q}} \bigg(\vert \Psi'(\varrho_{1})\vert^{q},\vert \Psi'(\varrho_{2})\vert^{q}\bigg). \end{align*}

Theorem 11. Suppose \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\) and \(\Psi:I^{\circ}\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\), \( \varrho_{1},\varrho_{2} \in I^{\circ}\) with \(\varrho_{1}< \varrho_{2}, q\geq1,\) and \(\Psi'\) is integrable function on the interval \([\varrho_{1},\varrho_{2}]\). Suppose \(\vert \Psi' \vert^{q}\) is an \(n\)-poly generalized convex function of Raina type on \(L[\varrho_{1},\varrho_{2}]\), then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{1-\frac{2}{q}}\bigg(\frac{1}{n}\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg]\bigg)^{\frac{1}{q}}A^{\frac{1}{q}} \bigg(\vert \Psi'(\varrho_{1})\vert^{q},\vert \Psi'(\varrho_{2})\vert^{q}\bigg), \end{align*} holds for \(\kappa\in[0,1]\), where A(.,.) is Arithmetic mean.

Proof. Suppose that \( \varrho_{1},\varrho_{2}\in I^{\circ}\). Since \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\), for any \(\kappa\in[0,1],\) we have \( \varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\in I^{\circ} \).

Assume that \(q>1\). From Lemma 1, power mean inequality, \(n\)-poly generalized convex function of Raina type of \(\vert \Psi' \vert^{q}\) and properties of modulus, we have

\begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\& \leq\bigg\vert\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\int_{0}^{1}(1-2\kappa)\Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))d\kappa\bigg\vert \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\int_{0}^{1}\vert1-2\kappa \vert d\kappa\bigg)^{1-\frac{1}{q}}\bigg(\int_{0}^{1}\vert1-2\kappa \vert \vert \Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert^{q}d\kappa\bigg)^{\frac{1}{q}} \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{1-\frac{1}{q}}\bigg(\int_{0}^{1}\vert1-2\kappa \vert\bigg[\frac{1}{n}\sum_{s=1}^{n}[1-(1-\kappa)^{s}]\vert \Psi'(\varrho_{1})\vert^{q}+\frac{1}{n}\sum_{s=1}^{n}[1-t^{s}]\vert \Psi'(\varrho_{2})\vert^{q}\bigg]d\kappa\bigg)^{\frac{1}{q}} \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{1-\frac{1}{q}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\int_{0}^{1}\vert1-2\kappa \vert\sum_{s=1}^{n}[1-(1-\kappa)^{s}]d\kappa +\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\int_{0}^{1}\vert1-2\kappa \vert\sum_{s=1}^{n}[1-\kappa^{s}]d\kappa\bigg)^{\frac{1}{q}} \\& =\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{1-\frac{2}{q}}\bigg(\frac{1}{n}\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg]\bigg)^{\frac{1}{q}}A^{\frac{1}{q}} \bigg(\vert \Psi'(\varrho_{1})\vert^{q},\vert \Psi'(\varrho_{2})\vert^{q}\bigg). \end{align*} For \(q=1\), we use the estimates from the proof of Theorem 9, which also follow step by step the above estimates. This completes the proof.

Corollary 9. If we put \(n=1\) in Theorem 11, then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\\;\;\;\;\;\;&\leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{4} A^{\frac{1}{q}}\bigg[\vert \Psi'(\varrho_{1})\vert^{q} , \vert \Psi'(\varrho_{2})\vert^{q} \bigg]. \end{align*}

Corollary 10. If we put \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 11, we get inequality (4.3) in [9].

Corollary 11. If we put \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 11, then we get Corollary 4 in [9].

Remark 9. Under the assumption of Theorem 11, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{E_{\alpha}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq \frac{E_{\alpha}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{1-\frac{2}{q}}\bigg(\frac{1}{n}\sum_{s=1}^{n}\bigg[\frac{2^{s}(s^{2}+s+2)-2}{(s+1)(s+2)2^{s+1}}\bigg]\bigg)^{\frac{1}{q}}A^{\frac{1}{q}} \bigg(\vert \Psi'(\varrho_{1})\vert^{q},\vert \Psi'(\varrho_{2})\vert^{q}\bigg). \end{align*}

Theorem 12. Suppose \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\) and \(\Psi:I^{\circ}\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\), \( \varrho_{1},\varrho_{2} \in I^{\circ}\) with \(\varrho_{1}< \varrho_{2}, q>1, \frac{1}{p}+\frac{1}{q}=1\) and \(\Psi'\) is integrable function on the interval \([\varrho_{1},\varrho_{2}]\). Suppose \(\vert \Psi' \vert^{q}\) is an \(n\)-poly generalized convex function of Raina type on \(L[\varrho_{1},\varrho_{2}]\), then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s}{2(s+2)}+\frac{\vert \Psi'(\varrho_{2 })\vert^{q}}{n}\sum_{s=1}^{n}\frac{s(s+3)}{2(s+1)(s+2)}\bigg)^{\frac{1}{q}} \\&\;\;\;\;\;\;\;\;\;+ \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s(s+3)}{2(s+1)(s+2)}+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s}{2(s+2)}\bigg)^{\frac{1}{q}}, \end{align*} holds for \(\kappa\in[0,1]\).

Proof. Suppose that \( \varrho_{1},\varrho_{2}\in I^{\circ}\). Since \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\), for any \(\kappa\in [0,1],\) we have \( \varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\in I^{\circ} \).

From Lemma 1, Hölder-Iscan integral inequality, \(n\)-poly generalized convex function of Raina type of \(\vert \Psi' \vert^{q}\) and properties of modulus, we have

\begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\int_{0}^{1}\vert1-2\kappa\vert \vert \Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert d\kappa \\& \leq \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\int_{0}^{1}(1-\kappa)\vert1-2\kappa \vert^{p}d\kappa\bigg)^{\frac{1}{p}}\bigg(\int_{0}^{1}(1-\kappa)\vert \Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert^{q}d\kappa\bigg)^{\frac{1}{q}} \\&\;\;\; + \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\int_{0}^{1}\kappa\vert1-2\kappa \vert^{p}d\kappa\bigg)^{\frac{1}{p}}\bigg(\int_{0}^{1}\kappa\vert \Psi'(\varrho_{2} +\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert^{q}d\kappa\bigg)^{\frac{1}{q}} \\& \leq \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}(1-\kappa)[1-(1-\kappa)^{s}]d\kappa +\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}(1-\kappa)[1-\kappa^{s}]d\kappa\bigg)^{\frac{1}{q}} \\&\;\;\;+ \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}\kappa[1-(1-\kappa)^{s}]d\kappa+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}\kappa[1-\kappa^{s}]dt\bigg)^{\frac{1}{q}} \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s}{2(s+2)}+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s(s+3)}{2(s+1)(s+2)}\bigg)^{\frac{1}{q}} \\&\;\;\;+ \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s(s+3)}{2(s+1)(s+2)}+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s}{2(s+2)}\bigg)^{\frac{1}{q}}. \end{align*} This completes the proof.

Corollary 12. If we put \(n=1\) in Theorem 12, then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{4}\bigg(\frac{1}{p+1}\bigg)^{\frac{1}{p}} \bigg[\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{3} +\frac{2\vert \Psi'(\varrho_{2})\vert^{q}}{3} \bigg)^{\frac{1}{q}}+\bigg(\frac{2\vert \Psi'(\varrho_{1})\vert^{q}}{3} +\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{3} \bigg)^{\frac{1}{q}}\bigg]. \end{align*}

Corollary 13. If we put \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 12, we get inequality (4.4) in [9].

Corollary 14. If we put \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 12, then we get Corollary 5 in [9].

Remark 10. Under the assumption of Theorem 12, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{E_{\alpha}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{E_{\alpha}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s}{2(s+2)}+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s(s+3)}{2(s+1)(s+2)}\bigg)^{\frac{1}{q}} \\&\;\;\;\;\;\;\;\;\;+ \frac{E_{\alpha}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2(p+1)}\bigg)^{\frac{1}{p}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s(s+3)}{2(s+1)(s+2)}+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\frac{s}{2(s+2)}\bigg)^{\frac{1}{q}}. \end{align*}

Theorem 13. Suppose \(I^{\circ}\) is an generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\) and \(\Psi:I^{\circ}\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\), \( \varrho_{1},\varrho_{2} \in I^{\circ}\) with \(\varrho_{1}< \varrho_{2}, q\geq1\) and suppose that \(\Psi'\in L[\varrho_{1},\varrho_{2}]\). If \(\vert \Psi' \vert^{q}\) is an \(n\)-poly generalized convex function of Raina type on \(L[\varrho_{1},\varrho_{2}]\), then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}k_{1} (s)+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}k_{2} (s)\bigg)^{\frac{1}{q}} \\&\;\;\;\;\;\;\;\;\;+ \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}k_{2} (s)+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}k_{1} (s)\bigg)^{\frac{1}{q}}, \end{align*} holds for \(\kappa\in[0,1]\), where \begin{align*} &k_{1}(s)=\int_{0}^{1}(1-\kappa)\vert1-2\kappa \vert [1-(1-\kappa)^{s}]d\kappa=\int_{0}^{1}\kappa\vert1-2\kappa \vert [1-\kappa^{s}]d\kappa=\frac{(s^{2}+s+2)2^{s}-2}{2^{s+2}(s+2)(s+3)}, \\ &k_{2}(s)=\int_{0}^{1}\kappa\vert1-2\kappa \vert [1-(1-\kappa)^{s}]d\kappa=\int_{0}^{1}(1-\kappa)\vert1-2\kappa\vert [1-\kappa^{s}]d\kappa=\frac{(s+5)(s^{2}+s+2)2^{s}-2}{2^{s+2}(s+1)(s+2)(s+3)}. \end{align*}

Proof. Suppose that \( \varrho_{1},\varrho_{2}\in I^{\circ}\). Since \(I^{\circ}\) is a generalized convex set with respect to \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}\), for any \(\kappa\in[0,1],\) we have \( \varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})\in I^{\circ} \).

Assume that \(q>1\). From Lemma 1, improved power-mean integral inequality, \(n\)-poly generalized convex function of Raina type of \(\vert \Psi' \vert^{q}\) and properties of modulus, we have

\begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\int_{0}^{1}\vert1-2\kappa\vert \vert \Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert d\kappa \\ &\leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\int_{0}^{1}(1-\kappa)\vert1-2\kappa \vert d\kappa\bigg)^{1-\frac{1}{q}}\bigg(\int_{0}^{1}(1-\kappa)\vert1-2\kappa \vert \vert \Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert^{q}d\kappa\bigg)^{\frac{1}{q}} \end{align*} \begin{align*}& \;\;\;+ \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\int_{0}^{1}(\kappa\vert1-2\kappa \vert d\kappa\bigg)^{1-\frac{1}{q}}\bigg(\int_{0}^{1}\kappa\vert1-2\kappa \vert \vert \Psi'(\varrho_{2}+\kappa\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))\vert^{q}d\kappa\bigg)^{\frac{1}{q}} \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{4}\bigg)^{1-\frac{1}{q}} \\&\;\;\;\times\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}(1-\kappa)\vert1-2\kappa \vert [1-(1-\kappa)^{s}]d\kappa+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}(1-\kappa)\vert1-2\kappa \vert [1-\kappa^{s}]d\kappa\bigg)^{\frac{1}{q}} \\&\;\;\; + \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{4}\bigg)^{1-\frac{1}{q}} \bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}\kappa\vert1-2\kappa \vert [1-(1-\kappa)^{s}]d\kappa+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\int_{0}^{1}\kappa\vert1-2\kappa \vert [1-\kappa^{s}]d\kappa\bigg)^{\frac{1}{q}} \\&\leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}} \bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{(s^{2}+s+2)2^{s}-2}{2^{s+2}(s+2)(s+3)} +\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\frac{(s+5)[(s^{2}+s+2)2^{s}-2]}{2^{s+2}(s+1)(s+2)(s+3)} \bigg)^{\frac{1}{q}} \\&\;\;\;+ \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}} \bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}\frac{(s+5)[(s^{2}+s+2)2^{s}-2]}{2^{s+2}(s+1)(s+2)(s+3)} +\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}\frac{(s^{2}+s+2)2^{s}-2}{2^{s+2}(s+2)(s+3)} \bigg)^{\frac{1}{q}} \\& \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}} \bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}k_{1} (s)+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}k_{2} (s)\bigg)^{\frac{1}{q}} \\&\;\;\;+ \frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}k_{2} (s)+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}k_{1} (s)\bigg)^{\frac{1}{q}}. \end{align*} For \(q=1\), we use the estimates from the proof of Theorem 9, which also follow step by step the above estimates. This completes the proof.

Corollary 15. If we put \(n=1\) in Theorem 13, then \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})}{8} \bigg[\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{4} +\frac{3\vert \Psi'(\varrho_{2})\vert^{q}}{4} \bigg)^{\frac{1}{q}}+\bigg(\frac{3\vert \Psi'(\varrho_{1})\vert^{q}}{4} +\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{4} \bigg)^{\frac{1}{q}}\bigg]. \end{align*}

Corollary 16. If we put \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 13, we get inequality (4.5) in [9].

Corollary 17. If we put \(n=1\) and \(\;{\mathcal{F}}_{\rho,\lambda}^{\sigma}(\varrho_{1}-\varrho_{2})=\varrho_{1}-\varrho_{2}\) in Theorem 13, then we get Corollary 6 in [9].

Remark 11. Under the assumption of Theorem 13, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} &\bigg\vert\frac{\Psi(\varrho_{2})+\Psi(\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2}))}{2}-\frac{1}{E_{\alpha}(\varrho_{1}-\varrho_{2})}\int_{\varrho_{2}}^{\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2})}\Psi(x)dx\bigg\vert \\&\;\;\;\;\;\; \leq\frac{E_{\alpha}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}k_{1} (s)+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}k_{2} (s)\bigg)^{\frac{1}{q}} \\&\;\;\;\;\;\;\;\;\;+ \frac{E_{\alpha}(\varrho_{1}-\varrho_{2})}{2}\bigg(\frac{1}{2}\bigg)^{2-\frac{2}{q}}\bigg(\frac{\vert \Psi'(\varrho_{1})\vert^{q}}{n}\sum_{s=1}^{n}k_{2} (s)+\frac{\vert \Psi'(\varrho_{2})\vert^{q}}{n}\sum_{s=1}^{n}k_{1} (s)\bigg)^{\frac{1}{q}}. \end{align*}

6. Applications

In this section, we recall the following special means of two positive number \(\varrho_{1}, \varrho_{2} \) with \(\varrho_{1}< \varrho_{2}\):

1. The arithmetic mean \begin{equation*} A=A(\varrho_{1},\varrho_{2})=\frac{\varrho_{1}+\varrho_{2}}{2}. \end{equation*}
2. The geometric mean \begin{equation*} G=G(\varrho_{1},\varrho_{2})=\sqrt{\varrho_{1}\varrho_{2}}. \end{equation*}
3. The harmonic mean \begin{equation*} H=H(\varrho_{1},\varrho_{2})=\frac{2\varrho_{1}\varrho_{2}}{\varrho_{1}+\varrho_{2}}. \end{equation*}

Proposition 1. Let \(0< \varrho_{1}< \varrho_{2}\), then

Proof. If \(\Psi(\varrho)=\varrho\) for \(\varrho>0\) in above Theorem 8, then we obtained the inequality (10).

Remark 12. Under the assumption of Proposition 1, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} \frac{1}{2}\bigg(\frac{n}{n+2^{-n}-1}\bigg)(\varrho_{2}+\frac{1}{2}E_{\alpha}(\varrho_{1}-\varrho_{2}))\leq\frac{E_{\alpha}(\varrho_{1}-\varrho_{2})+2\varrho_{2})}{2} \leq A(\varrho_{1},\varrho_{2})\frac{2}{n}\sum_{s=1}^{n}\frac{s}{s+1}. \end{align*}

Proposition 2. Let \(\varrho_{1}, \varrho_{2}\in (0,1]\) with \(\varrho_{1}< \varrho_{2}\), then

Proof. If \(\Psi(\varrho)=-\ln \varrho\) for \(x\in (0,1]\) in above Theorem 8, then we obtained the inequality (11).

Remark 13. Under the assumption of Proposition 2, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} \frac{1}{2}\bigg(\frac{n}{n+2^{-n}-1}\bigg)\ln G(\varrho_{1},\varrho_{2})\leq \frac{1}{\varrho_{2}(\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2}))} \leq \ln(\varrho_{2}+\frac{1}{2}E_{\alpha}(\varrho_{1}-\varrho_{2}))\frac{1}{n}\sum_{s=1}^{n}\frac{s}{s+1}. \end{align*}

Proposition 3. Let \(0< \varrho_{1}< \varrho_{2}\), then

Proof. If \(\Psi(\varrho)=\frac{1}{\varrho^{2}}\) for \(\varrho\in \mathbb{R}\setminus\{0\}\) in above Theorem 8, then we obtained the inequality (12).

Remark 14. Under the assumption of Proposition 3, if we take \(\sigma=(1,1,...)\) with \(\rho=\alpha,\; \lambda=1\), we get the following inequality involving classical Mittag-Leffler function \begin{align*} \frac{1}{2}\bigg(\frac{n}{n+2^{-n}-1}\bigg)\frac{1}{(\varrho_{2}+\frac{1}{2}E_{\alpha}(\varrho_{1}-\varrho_{2}))^{2}}\leq \frac{1}{ \varrho_{2}(\varrho_{2}+E_{\alpha}(\varrho_{1}-\varrho_{2}))} \leq \frac{1}{H(\varrho_{1}^{2},\varrho_{2}^{2})}\frac{2}{n}\sum_{s=1}^{n}\frac{s}{s+1}. \end{align*}

7. Conclusion

In this paper, we have introduced a new family of convex functions namely \(n\)-poly generalized convex functions of Raina type. We established a new version of Hermite-Hadamard type inequality and some of its refinements. We believe that this new family of convex functions will have very immeasurable and chasmic research in this mesmerizing and absorbing field of inequalities and will inspire interested readers.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Energy policy, in general, is the stipulation of standard procedures and regulations by governmental or global bodies to look after the issues related to energy generation, distribution and end-user consumption. The salient features of energy policy include local or global legislations, treaties, investment-incentives, regulations regarding conservation of energy, monetary tax regulations and other significant factors impacting public well-being and development. In the current era, energy acts as the pivot to all economic activities of a region in general and a country in particular as it is required by the industry, agriculture, communication and transportation sectors among several others.

With the development and implementation of modern renewable energy systems at global scale, the dimensions of energy policy have evolved into renewable energy policy. The renewable energy policies are not only being developed and implemented at state level through governments, but also at global level through international agencies, syndicates and think tanks like IRENA or the International Renewable Energy Agency, REN-21 or the Renewable Energy Policy Network for the 21st Century, REEEP or the Renewable Energy and Energy Efficiency Partnership, ISA or the International Solar Alliance, EREC or the European Renewable Energy Council and the Energy and Climate Partnership of the Americas (ECPA) to name a few.

The renewable energy policies take into account deployment planning, legislation on commercial aspects of renewable energy systems along with their efficiency standards, fiscal policies, energy security and implementation of the above in accordance with international treaties, agreements and alliances. Another major issue being looked after in the global policy making is the effect of all these systems upon the climate, environment and ecology [1].

The major factors which are defined in the regional or global renewable energy policies include;

1. Present and future potential of self-sufficiency with regard to renewables,
2. How and where the renewable systems will be deployed,
3. Management of renewable energy distribution,
4. Environmental impact of the deployed renewable energy setups,
5. Encouragement of energy efficient hardware adoption,
6. Technology Modernization and reduction in Green House Gas (GHG) emissions,
7. Incentives and mechanisms to promote the use of renewables at regional and global scales [2].

Effective policies on renewable energy systems can give an impetus to innovation in manufacturing and cost reduction along with their adoption as a clean and cost-effective source of power generation. Different countries have adhered to different laws and renewable energy policies to promote the adoption of renewable energy technologies [3]. The significance of these policies usually varies from one country to the other and is primarily based on their geographical diversity, economic conditions, preferences and social trends.

Global policies to propagate renewable energy have played a key role in their expansion. Europe is regarded as the pioneer in defining renewable energy policy in early 2000s. Following the footsteps of Europe, now most countries around the world have formed such policies to cater for their plans to deploy and promote renewable energy systems. Major inter-governmental organizations are playing a key role for promoting the adoption of renewable energy worldwide. They have been providing policy advice and motivating member countries to develop and improve renewable energy policies apart from helping them with capacity building and technology transfer [4]. These organizations have also identified the fact that the renewable energy has the potential to take poor countries to a higher level of prosperity.

The global policy on renewable energy systems is also having a positive impact on evolution of next generation technologies. As the focus is now shifting from fossil fuels to renewables, several new technologies have been created as a result of more focused research and development in the renewables domain. These newly evolved technologies are not only proving to be cost-effective, but also offer an easier, effective and efficient way of their deployment regardless of where they are being deployed.

A close study of literature related to global renewable policy brings to light the fact that the momentum of fossil-fuel market and consumption still impedes the adoption of renewable energy systems owing to the geopolitical preferences of key global players. Some of the countries have even adhered to the policy of discouraging the proliferation of renewable energy systems not just within their own political boundaries, but also in their neighboring regions and trade-partners. Despite these impeding factors, effective policy-making vis a vis renewables is still on-going at national as well as global scale and is gradually gaining the pace to address all the issues related to the growth of renewables for sustainable development and complete reliance on green energy resources.

This paper will encompass all the key policies which are being introduced and implemented by inter-governmental organizations across the globe and will also highlight the policy preferences and structures being adopted by various countries across different regions regarding the development and promotion of renewable energy resources. It will also discuss which renewable policy is lopsided in the favour of which renewable technology by using the detailed data provided in the latest global renewable energy policy reports recently published in different research studies critically analyzing different governmental policy instruments.

2. Literature Review

Policies pertaining to renewable energy systems (RES) play an important role in their effective deployment and promotion for any country. Due to diminishing fossil fuel reserves, their inherent tendency of spreading the much talked about greenhouse gases (GHGs) and increasing prices, nearly all countries across the globe have formulated some type of policy regarding the promotion and deployment of renewables to counter these adverse factors. Policies of some developed countries or key players in the domain of RES are exemplary while for some, the policy formulation process either has some loopholes or is still in its nascent stages.

There are several publications regarding the comparative analyses of renewable energy policies which have highlighted the pros and cons of these policies either by directly discussing their quality or effectiveness in short-term, as well as long-term scenarios or by comparing the magnitude of their quality as compared to policies of other countries or those defined by inter-governmental organizations on renewables. Having studied the reports, journals and technical papers on global renewable energy policies, one can easily conclude that the policy formulation and their implementation are still not perfectly defined and are constantly evolving with several modifications and improvements with every passing year. Some of the countries which have achieved the technical excellence in manufacturing and deployment of renewable energy systems through dedicated research and development are in a position to take effective policy measures Vis a Vis their implementation and effectiveness. Some policies are unique to a particular country or region while some are common across the globe. These policy matters are significantly affected by social preferences, political tendencies and geo-political scenarios.

Most of the studies begin with discussing the current issues in the domain of renewable energy with reference to policies of the past and those which have been currently implemented. Then a solution in the form of a new business or technical model is suggested and a restructured policy is proposed. In [5], S. Han et al., have discussed the current trend of renewable energy policy in South Korea. They have discussed the contemporary issues associated with instability in the supply of power from renewable energy sources along with the low efficiency issues offered by such systems. By observing the national policy trends, they have tried to propose an effective business model which could be used to address afore mentioned issues. They have also identified that the development and implementation of renewable energy systems is not the only solution in this regard, rather there is a need to develop a compact and interactive power system integrated with state-of-the-art information and communications technology (ICT) based solutions and platforms. Another observation made in [5] is that efficient energy use could be achieved by deploying advanced energy storage systems (ESSs) with an intelligent and agile demand-side management systems such as the EMS or the energy management system also termed in the modern technological nomenclature as the demand response (DR) system.

In one study about electrification of Ethiopian Power Sector using renewables, Md Alam Hossain et al., [6] suggest the use of diversified energy supply-mix to achieve energy security and sustainable energy system development with zero carbon emission and increased export options. It also suggests the use of market allocation (MARKAL) energy system model of energy to achieve all these mentioned objectives. For diversification of renewable energy sources, a mixed use of frontline renewable energy technologies like Hydro, PV and wind have been suggested. The effectiveness of Market allocation scheme has also been discussed by Ricardo Fagiani et al., [7] wherein such a policy of the European Commission implemented to introduce competitive allocation mechanism has been discussed. This policy ensures that the market players reveal their actual generation cost which would further help the policy makers with effective price formulation. It also discusses the pros and cons of commission's recommendations regarding allocation of feed-in premiums (FiPs) through tender and quota obligation schemes. In [8], Song Yun-Wei has examined the renewable energy policy development in the USA. The policies regarding the green consumption and consumer protection have been discussed at stretch to verify their effectiveness. He concludes his study with the argument that renewables still suffer to compete with traditional energy due to the high cost of its implementation. He also states that the renewable technologies are still under development for their efficiency and reliability and that the public is still unaware of the benefits of deploying renewable energy harnessing techniques. Other obstacles that have been identified include the lack of research and development funds and the subsidies that still exist for the deployment of power generation systems using traditional non-renewable sources.

The single largest barrier in the non-adoption of renewable energy systems globally is policymaking as identified by Nathan Murthy et al., [9]. They propose that effective policy recommendations are needed to accelerate the penetration of renewables at national as well as global scale. Their study also states that a sustained policy support for demand dispatch in combination with integrated intermittent generation through renewables is important to achieve the goal of meaningful inclusion of renewables to the grid. It also been suggested that a National Action Plan be proposed to delineate a comprehensive climate strategy, incentives be introduced to promote renewables against the traditional fossil fuel-based generation systems, subsidies to fossil fuel-based producers be removed in favor of renewables and an effective Feed-in Tariff (FiT) policy be introduced. A study about the renewables policy in different States of the US has been presented by Olawale Ogunrinde et al., [10] . According to their report, several policies to incentivize the proliferation of RESs have been implemented in several states. One of these policies is the renewable portfolio standards (RPS) which ensures that renewable energy sources contribute to a certain percentage of the total electricity sales. It has been observed that the returns on RPS are location specific and RPS mandate implementation is still not fully functionalized due to lack in development of renewables integration technologies.

China's renewable policies and regulations have been discussed in [11] which suggests that China has a strong policy structure to achieve the sustainable development and to overcome the menace of climate change. According to this report China has implemented instruments from laws, policies and financial incentives. Chinese policymakers have clearly identified that renewable energy is without a doubt unavoidable for sustainable economic growth and development. They have been weaving their policies around the idea that it is imperative to have reduced production cost and technological reliability of renewable energy systems. To further the effectiveness of China's policies regarding renewables, authors suggest that legal and institutional frameworks along with a supportive policy must be brought forth. Security and sustainability of public sector, promoting the involvement of private sector, R&D across all sectors, establishment of small and medium enterprises related to renewable energy systems and access to affordable finance have been identified as areas which must be taken into account for improvement in existing policies. Reforms are also needed create a positive investment climate so that the attention of private sector with its capital towards renewable energy could be attracted which will further expand international cooperation in this domain. Further policy suggestions in [12] include development of advanced and efficient renewable energy systems which could be achieved with the definition of specific energy development objectives, mandatory and effective legislation, administrative intervention and establishment of a complete research and design institution.

Shyam B. et al., in [13] discuss policies, opportunities and challenges related to renewable energy systems in India. India is one of the global leaders in the deployment of renewable energy systems and has implemented an effective policy mechanism to promote the deployment and utilization of renewables across the country. It is known to Indian policy makers that utilization of renewable energy is crucial to nation's energy security and economic stability. In India, the renewable energy sector is supported by the government with significant amount of financial, educational and institutional aids. Owing to India's effective and reliable renewable policies, strategies and frameworks, the country has witnessed prolific deployments of renewable energy harvesting systems widely dispersed across the country. The publication also discusses the potential challenges faced by the power sector in integrating the renewable energy systems with the existing grids and significance of smart-grid in this regard. It also highlights the work being done in the domain of smart-grid to overcome all the challenges related to renewable systems integration and energy management. Another important take-away from this discussion is the healthy environment created by the government to ensure the involvement of private sector, particularly in the domain of wind power. R&D in the domains of micro-grids, storage systems and smart-grid is being emphasized. Authors in [14] introduce the generation expansion planning (GEP) method and policy and evaluate its effectiveness. The objective of GEP is to identify the most feasible combination of different conventional or fossil-fuel based energy sources in combination with RESs to overcome the power demand in a predictable manner. It involves a clearly defined algorithm for its implementation which monitors all the demand and available supply parameters for optimized decision making strictly including parameters from renewable energy sources. Results from this publication conclude that committed higher percentage of renewable energy portfolio (REP) results in a comparatively higher investment and somewhat lower cost of operation. It also concludes that the policy makers make use of GEP to identify the most efficient and economically feasible policy while also determining the most suitable renewable portfolio standard (RPS) or emission penalties.

Visible and effective leadership and commitment by high level policy and decision makers is important for significant policies regarding the renewables [15]. Policies to stimulate the growth of industries that manufacture, supply and trade RESs with solutions must be introduced. Strong and stable financial services and institutions for private and public investments must be implemented which have high transparency and accountability [15]. Many publications discuss the significance of penalties, FIT, FIP and RPS in policy making and their repercussions Vis a Vis renewable energy systems efficient deployment and acceptance in the current era [16]. Narayanan et al., in [17] discuss and propose the significance and inclusion of micro renewable energy systems (MRES) in the future policies to maximize the utilization of renewables. According to [18], strict quota rules are needed for renewable energy development as well as punitive measures are needed for those who do not achieve the production and dispatch targets. Effective communication is needed between the state-grid and renewable energy generators. A detailed literature about latest renewable policies in the US and EU [19] compares and discusses the effective renewable policies which have been formulated and implemented in these regions. The discussion again highlights the significance of smart-grid based energy management, monitoring and storage systems to achieve the goal of efficiency in modern micro and large-scale renewable energy systems. The discussion in [19] describes the significance of renewable energy certificates (RECs) and FiTs which are being widely used in modern renewable policies, particularly in the policies formulated by the key global players in renewable energy systems and their implementations. According to this discussion the RECs ensure maximum possible inclusion of RESs by allocating compulsory or committed generation capacities. It also introduces and states the significance of RPS and solar renewable energy certificates (SRECs). The discussion concludes with remarks that the EU and the US have introduced micro-generation and smart-grid initiatives in their policies. Further, incentives for private sector have been proposed to ensure their participation in the research, manufacturing and deployment domains of RETs. In [20], renewable policies of different countries have been discussed and their salient features have been described and compared. The common targets of these policies include increase in renewable energy, increase in renewable efficiency, energy security and sustainability, competition in renewable energy generation and sustained environmental protection.

In [21], the significance of RPS, Production Tax credits, FiTs and net-metering in renewable policies have been discussed in the USA and its states. The discussion in [22] stresses upon the restructuring of financial models in renewable policies to maximize the investment potential of the interested parties in the market. This would help create a healthy environment favoring the growth and investment in RETs. Enerallt, a market based inter-region energy system model, has been proposed in [23] in an attempt to model a common energy market based on RETs, implementation of which would require a policy shift by the participating countries. Authors of [24] discuss traditional renewable policies at a stretch and identify and propose areas where new policy formulation or improvement is required. These proposed areas for policy formulation and improvement include RPS, Net-metering policy, renewable energy certificate (REC), electricity feed-in law, public benefit funds, investment support, fiscal and financial measures, tenders and quotas, emissions trading policies and renewable energy targets.

Nearly all of the previous works discussed here state many common improvement factors in the renewable policies being formulated and implemented around the globe. These also discuss the existing challenges along with areas having significant room for improvement as far as policy formulation or policy restructuring is concerned. Many new business, financial and technological models have also been proposed which could be used globally to implement an effective and sound policy related to renewables.

3. Renewable Energy Development and Implementation around the Globe

Renewable Energy policies vary globally but share the common objective of achieving the deployment and promotion of renewable energy technologies at the least cost and minimum damage to the local and global environment. Elements commonly shared by all these policies include a specific target, an annual target, a set of feasible and eligible renewable technologies (RETs), policies and considerations regarding the import of equipment related to RETs and an effective compliance and enforcement structure. This section discusses modern renewable policies which are evolving with every passing day for feasible and effective adoption and implementation of RETs for self-sustained, secure and environmentally friendly power generation systems.

3.1. Policies to Integrate ICT Technologies with RETs

To achieve the goals of efficient power generation, transmission, distribution and end-use, global renewable policy formulation regimes are earnestly looking forward to introducing policies which ensure integration of information and communication technology (ICT) with newly deployed as well as existing grid infrastructure. The prime example of implementation of ICT in the domain of renewable energy systems is in the form of smart grid. Smart grid systems employ various ICT based mechanisms to enhance power quality and stability particularly where intermittent renewable energy sources contribute to the electricity grid.

Renewable Energy policies vary globally but share the common objective of achieving the deployment and promotion of renewable energy technologies at the least cost and minimum damage to the local and global environment. Elements commonly shared by all these policies include a specific target, an annual target, a set of feasible and eligible renewable technologies (RETs), policies and considerations regarding the import of equipment related to RETs and an effective compliance and enforcement structure. This section discusses modern renewable policies which are evolving with every passing day for feasible and effective adoption and implementation of RETs for self-sustained, secure and environmentally friendly power generation systems.

Smart grid technologies make use of ICT for gathering information related to generation transmission and distribution performances and also for actuating processes that make the overall system resilient with enhanced power quality, sustainability and economy with regard to the power systems.

The evolving smart grid technology brings along with it a novel concept related to demand-side response and management system, known as the Electric Spring, to achieve high standards of power quality and overall distribution system stability with the introduction of intelligent and interactive smart loads to the existing power supply infrastructure [25]. Modern renewables policy making approach must focus on this concept as the smart loads cater for nondeterministic or fluctuating power generation by dynamically varying the load demand, specifically for non-critical loads. Smart loads are in essence power reactive power controllers based on modern power electronics, integrated with non-critical end-use devices or loads. Secondly, such a system is also proposed to be capable of self-regulating the voltage values at the distribution-side where intermittent generation sources are connected. Introduction of such systems as per policy would help mitigate the issue of voltage fluctuation particularly in areas where intermittent RESs are connected to the grid. Smart loads are expected to be capable of dynamically injecting reactive power to the distribution-side network to achieve localized support for voltage adjustment [25].

In order to streamline the integration of renewable technologies to the grid, the use of smart grid technology has increased manifolds. Utilities are increasingly making use of them to ensure optimum performance and uninterruptible delivery. The current urbanization trends demand for increased use of renewable technologies which are now contributing towards the efficient demand and supply management. This increased utilization of renewable energy sources has also introduced risks related to voltage and power fluctuations, harmonics, frequency mismatch and inefficiencies in generation, transmission and distribution. With the introduction of policies and solutions using smart grid technologies, small distributed energy sources can be integrated to an urban power supply network allowing real-time management and optimization of such integrated systems with the existing infrastructure. Such solutions are bound to play an essential role in sustainable development of smart cities along with other ventures aiming at establishment of self-sustaining and self-sufficient energy generation and distribution systems. Figure 1 shows a scalable model for smart grid-based setup with renewable energy sources integrated with the primary grid [26].

Another direction, again based on ICT in general and smart grid in particular is the policy-based introduction of smart city concept amongst the masses and relevant industries. A Smart city comprises of several components which are core to its concept as a future efficient and cost-cutting technology utilizing the best features of smart grid technology. These components include effective governance, mobility of assets and contributing resources, economy and efficient energy harnessing and subsequent power distribution practices. These components could play a key role in achieving a sustainable urban life, integrating power generation, distribution and end-use infrastructure along with various contributing stakeholders. Smart cities are considered to be a logically extended concept of smart grid technology and its practical implementation is of utmost importance to the modernization process of the traditional power systems. Policy formulation related to the introduction of smart grid systems in general and smart city concept in particular must be focused on grid integration of renewable energy resources, agile, stable, efficient and capacious energy storage systems, smart lighting systems and electric vehicles for a green global environment, energy security and sustainability [27].

Policy making is also bound to be focused on the research, development and introduction of new power electronic devices capable of either taking remote commands using communications infrastructure or making intelligent decisions in real-time to manage the power systems for its smooth performance. These devices are being used to integrate distributed generation (DG) or renewable power generation systems at the main supply with energy storage systems (ESSs) working in parallel, particularly in the case of solar and wind-based power generation setups. In conjunction with network infrastructure and smart meters, these devices can be used for power grid automation and control. These devices can also play an important role for grid infrastructure security, effective demand response and power quality [28].

Integration of ICT technologies to RETs has been adopted as a core part of new renewable policies around the globe, particularly by the countries of the developed world. The introduction of these policies is still in somewhat early stages yet their introduction has started providing substantial dividends to those countries in the form of secure, stable sustainable, efficient and effectively manageable power infrastructure with an ever-increasing chunk of renewable energy systems contributing to the overall power infrastructure. Introduction of these policies has also started to help achieve green environment with minimal harm to natural and ecological processes. Further research and development as a part of policy will accelerate the adoption rate of this technology with increased sustainable development and better global environment.

3.2. Policies to introduce new business, financial and planning models for effective adoption of renewable energy systems

Business models pertaining to any global or regional market play a key role in adoption or promotion of a particular technology. If the business model is sufficiently effective, industries and masses can be compelled to adopt a technology which, according to technologists and policy makers, is good for the community and industry but is not experiencing substantial adoption.

In order to overcome high initial investment costs, environmental constraints impeding the deployment of RET based systems, commercialization difficulties, issues related to deployment, operations and maintenance and the barriers including unstable power supply and low system efficiencies, an integrated system of diverse energy sources, as discussed in the previous sub-section, is needed along with a new business model [5]. These issues cannot be overcome without government support and require a clearly defined business and financial models designed initially formulated and implemented by the government with sufficient incentives and guarantees to lure in investors and stakeholders. Several governments are working in the direction of discovering an optimum business model for their indigenous as well as regional renewables market to accelerate the promotion and adoption of RET based systems. One such model is the ICT model, being introduced in the developed world to create energy efficient power infrastructure fully integrated with multiple RET based power sources. The efficiency offered by such systems itself is an attraction for the investors and other new entrants to the renewables market.

Another renewables expansion model is the generation expansion planning (GEP) model. The purpose of GEP is to identify the most economically viable and optimal combination of traditional and renewable energy sources to accommodate for predicted power demand [14].

Micro Renewable Energy Systems (MRES) deployment model is another dimension which could be opted by policy makers not only to encourage the use of RETs but also to meet the ever-increasing demand of power in a distributed manner. MRES involves RET based power generation units of less than 5 kWh capacity. These systems are either community owned or individually maintained to fulfill the daily requirement of scattered or distributed communities. The main aim of using MRES is to reduce reliance on traditional non-renewable systems which further helps reduce the harmful impact of fossil-fuel based systems with regard to the greenhouse gases (GHGs) [17]. With effective policies in support of MRES model in place, micro renewable energy devices will become popular enough to be readily bought by the people allowing energy independence.

The problems currently being faced by such micro renewable energy systems is the lack of their commercial presence and inefficiency of such systems. These problems can be significantly overcome once the MRES model comes into action as this would also bring along improvements in such systems with broader commercial presence. Governments and their policy making bodies must direct funding towards R&D of such systems with subsidies, incentives and guarantees to ensure affordability and deeper penetration of such systems to the mainstream renewables market. Table 1 shows the advantages that come along the implementation of MRES model [17].

As far as investment and financial models are concerned, they face a few constraints in the form of lack of innovation in the RET systems deployment methodologies and investment climate. The innovation model itself involves economic risks, political climate dependence, natural risks, social acceptance, governance issues and natural factors determining its effectiveness. Innovative models and projects depend upon initial costs, time required for market maturity and time required to bring the innovation to the market. Innovation is strictly involved in the maturity and stability of the investment and financial models. Next to the innovation model, there are two further finance models which require policy-makers attention. These include (i) program-targeted financing model and (ii) market-based financing model. These two models differ on the basis of their objectives and financing structures and sources.

In the case of program-targeted financing model of RES development, government gets directly involved in funding implementing non-debt-based funding structure. This model may also include the involvement of government-owned enterprises in partnership with private investors. The funding in this scenario is more targeted towards domestic socio-economic progress and self-sufficiency in RET based power generation. In the market-based financing model, general market mechanics work as the core driving factors like demand and supply trends, profit earning and sharing mechanisms and risk reduction practices. Unlike the program-targeted financing model, this model focuses on export of energy on region-wide scale. The program-targeted model may prove beneficial for the developing or cash-starved nations while the market-based financing model is likely to work better with developed nations. Both these models can prove beneficial for the development of alternative energy sources and should be the focus of renewable energy policy makers. These models would help enhance the energy potential of industries, create suitable environment for increased investments in the renewables domain and would help create more competitive market base. Having these models implemented would most likely ensure self-sufficiency, energy security and socio-economic well-being.

Table 1. Advantages of MRES model.

Advantage	Impact
Self-sufficiency in Power generation	Savings on utility bills and additional source of income
Community/individual ownership and participation in RET based systems deployment	Higher approval of RET based systems adoption and job creation
Localized power generation	Energy security and reduced cost of logistics
A new technological domain with improvised generation methods	An impetus to technological advancement and newer sources of income
Independence from energy market fluctuations and volatility	Reduced bills with higher system stability

3.3. Policy recommendations to ensure deep penetration of RET based systems

Policy making with regard to renewable energy systems has witnessed significant growth and maturity over the past decade. Many countries have evolved different policy models to ensure the penetration of renewable energy systems to the national and regional markets. A similar role is being played by the global renewable policy making regimes to permeate the policy effects across the globe so that the growth of RET based systems is uniformly experienced by the global stakeholders. There are several recommendations which are either in place or are being considered by different countries for the promotion of renewable energy systems. Some of the policy recommendations are in nascent stages and need to be earnestly looked at by the policy makers to achieve the desired objectives.

3.3.1. National Energy Action Plan:

Such action plans need to be adopted to ensure formulation and implementation of a stable renewables policy so that the significance of RET could be realized at national level. This would also help reshape industries in accordance with renewable policies and would help renewables market to evolve in the mainstream.

3.3.2. Production Tax Credits (PTCs):

Production Tax Credits must be offered to the manufacturers of RES related devices. Demand Response (DR) management providers should also be offered such credits. These credits work as an incentive for the promotion of renewable systems and gradually discourages the use of traditional fossil-fuel based generation systems. PTCs will attract private investment and will minimize the investment risks often faced during the implementation of newly evolving technologies.

3.3.3. Reduction or removal of subsidies:

Subsidies which are being offered to the power producers using traditional fossil-fuel based technologies should either be gradually reduced or be removed altogether. This move would definitely help masses and industries alike in opting for renewable generation systems to fulfil their energy requirements while conserving the natural environment and climate.

3.3.4. Policies related to Feed-in Tariff (FIT):

A balanced and attractive feed-in tariff policy must be formulated, based on the type of renewable energy technology used, to encourage the independent or individual renewable energy suppliers so that the contributors get rewarded for the amount of power they are injecting into the power supply system. The FIT's mechanism is such that the government agencies mandate the distribution companies to purchase power generated by RE based generators at a fixed or pre-decided price over an extended period. This allows RE based power companies to make investments which are less risk-prone by allowing them to have a known fixed retail price of electricity. This property of FIT mechanism has allowed it to be more popular way of contributing with the RE based power [9]. FIT is often regarded as an incentive which has been implemented by many countries across the globe. FIT and its related policies encourage small businesses to deploy and maintain RE sources. The main factors that these tariffs include guaranteed and committed grid access, power trade at market-based price, long-term power purchase agreements, [19].

3.3.5. Policy related to Renewable Policy Standard (RPS):

The RPS ensures that the power being contributed to the electricity supply system using RET costs more than the standard wholesale price of the electricity. It not only rewards the suppliers with the amount of power they are injecting but also helps them get their due share for the maintenance and up keeping of the renewable energy harnessing assets they have deployed [9]. The RPS comprises of the following key elements: (i) setting the mandatory quota of power generated using RES as stipulated by the government agencies; (ii) issuance of Tradable Green Certificates (TGCs) to RES based power suppliers based on their respective contribution; (iii) selling and purchasing of predefined quota certificates between stipulated targets and RE based suppliers. Setting of such mandatory targets helps adopt RE based set-ups at low cost resulting in high adoption trend of such systems. The main difference between the FiT and RPS is that the FiT is purchase price oriented while RPS is installed capacity oriented. In general, price regulations apply on FiT and quantitative regulations apply on RPS [16]. If FiT and RPS are compared for their effectiveness, the former has been found to be more productive in proliferating the RET based systems owing to its popularity and capability to help inject more RE based power into the system. Despite the popularity of FIT based systems, RPS system can be further improved to help the growth of RE based systems deployment. One drawback found with the FIT is that it causes the price of deployment and devices to increase manifolds. RPS based system, if implemented with a few improvements, may help create a balance between the adoption of all types of RET based systems keeping device and deployment costs in check [16]. RPS is basically designed to maintain a mix of different RESs hence giving the opportunity to less-adopted Renewable technologies to evolve further [19].

3.3.6. Ecotax:

Ecotax system as a part of policy allows policy makers and concerned government agencies to give tax incentives to entities that build or are interested in deploying RE based sources. Such a system is most effective when those individuals, utilities or power distributors are penalized which contribute to pollution and harm to the environment [19].

3.3.7. Renewable Energy Credits (RECs):

Renewable energy credits ensure maximum adoption of RE based systems by assigning mandatory generation capacities. RECs can be traded between companies to guarantee generation using RET based systems [19].

3.4. Formulation of effective renewable energy laws

Energy laws play an important role to promote the development and adoption of RESs. Energy laws are designed to enhance the energy efficiency and help conserve the energy by introducing different legal regulatory measures. Laws, particularly with regard to renewable energy systems, help formulate economic incentives and energy control in an effective and flexible manner [8]. In order to develop and promote RES, consistent policies are needed. For RES to flourish on a national level, it is important that the investors, developers, researchers and end-users find it viable to make decisions on long-term basis as far as adoption of RETs is concerned [8]. The renewable energy law comprises of a policy framework which help establish key policy ingredients like national RE targets, compulsory connection and purchase policy, a FIT system according to national priority, strategies for funding and financing of RE incentives. The primary objective of such laws is to save energy and protect the deteriorating environment [18]. Introduction of new and improved energy laws have been found to be capable of expediting the development and implementation RETs at a large scale.

The key mechanisms that contribute to the effective renewable energy law are as follows;

1. Clear and elaborate definition of renewable energy or green energy sources. The national target should be prioritized utilization of these sources to meet energy demands.
2. Governmental agencies looking after energy affairs must be made responsible for planning and management of renewable energy. If need be, dedicated agencies must be formed that work in conjunction with the existing agencies.
3. Transmission and dispatch companies should sign agreements with RE based generators to purchase the generated power and provide related services for the ease of their functioning.
4. A targeted percentage of RE must be stipulated in the newly formulated laws to ensure that a mandatory chunk of energy comes from RE sources.
5. The central government must be capable of looking after the law implementation and must intervene wherever policy related loopholes or legal misconduct is found.
6. Special funding regimes must be formed, the sole purpose of which should be to support the research and development of efficient and cost-effective RETs [18].

4. Global Renewable Policy Implementation and Comparison

This section of the paper reviews policies and energy strategies of a few countries, most of which are global leaders in successful development and implementation of RETs. It also encompasses policy-based measures being formulated for environmental protection and proliferation of RE based systems.

The world has been witnessing a steep rise in energy demand to achieve the goals of development not just at industrial scale but also at the level of masses. People around the globe have started realizing the importance of renewable energy technologies and energy conservation as these contribute not just towards environmental protection but also towards the creation of new avenues and dimensions of economics. Of all the RE sources and technologies available, the most sought-after technologies and sources have been solar, wind and hydropower. With evolving policies Vis a Vis RE sources, the world has experienced sharp increase in the use of almost all forms of RE sources as it has been realized that the use of RETs is necessary to achieve self-reliance, energy security and environmental protection. Renewable energy policies of many countries have contributed to growth in the market volume of RE systems and also to the increase global trend of adoption for such systems. In the pursuit of efficient RE system, policies of many countries have helped evolve new technologies which complement the use of RE to a great extent. These new technologies or approaches include hydrogen fuel-cell based storage systems, ICT and nano-technology based power management and control systems. Even though many countries have effective policies in place regarding RES, there still remains enough room for improvement of such policies to make them more effective.

4.1. Renewable Energy Policies of Australia

The Australian government, in several of its energy policy frameworks, has set targets of achieving secure, reliable, green and low-cost energy to domestic and industrial consumers in Australia. Another objective that was clearly defined was the attainability of sustainable development of national energy sources. The government also decided to create and maintain a regulated and competitive energy market for the propagation of energy harnessing using cost-effective, efficient and clean technologies. Energy efficiency and security being the primary objectives of the Australian government and policy makers, the country has established several new RE based generation systems with large energy storage projects, the most popular being the 129 MWh Hornsdale Power Reserve Complex in Southern Australia, complementing the working of 315 MW Hornsdale windfarm project.

4.2. Renewable Energy Policies of Brazil

The Brazilian government is mainly targeting energy efficiency and market liberalization with regard to RE systems. The focus for establishment of RE systems is not just to entertain the residential consumers but also the Industrial consumers. Brazil being one of the major producers of clean and green RE in the world owing to the presence of several hydel sources, is focusing on biofuel-based projects owing to the presence of Amazon Rain Forest on its mainland which is capable of delivering huge amounts of biomass for that purpose.

4.3. Renewable Energy Policies of China

Owing to effective new policies regarding renewable energy, China is undoubtedly the single largest renewable energy producer. China is taking the issue of climate change seriously and has put in place several policies regarding conservation of ecology and environment for sustainable development and better socio-economic future of the country. The National Development and Reform Commission of China, in 2012, introduced the pilot carbon cap and trade scheme to minimize the carbon emission to the environment for a cleaner and greener future. These effective policies have made China the leading user of PV, hydro and wind based RE systems [20].

4.4. Renewable Energy Policies of EU

EU has set several new targets through legislation to achieve energy security and climate conservation using renewable energy. They EU's policy makers have put forth several proposals to ensure the availability of a green energy harnessing environment through several policy proposals, in particular the 20/20.20, like;

1. 20% Reduction in the emission of greenhouses gases (GHGs),
2. 20% increase in the overall energy efficiency,
3. Increase the contribution of renewable energy by 20%,
4. Increase the contribution of biofuels for transport by 10%.

EU, as a single contributor to the renewable energy domain, is one of the leading RE producers. Effective policies of the EU have allowed common individuals and small industries to contribute renewable energy, often in surplus [20].

4.5. Renewable Energy Policies of India

India's energy needs are driven by its huge population living without access to electricity. With huge burden on other energy sources, the need has driven the Indian government to formulate highly effective renewable energy policy to make energy available for all, regardless of geographical location. India has now become one of the leading users of biofuels, wind and solar power due to highly focused policy making of the Indian government. One such policy is India's Integrated Energy Policy (IEP) which has helped meeting energy demands, ensured secure energy supply, protection of the environment and deeper penetration of RET based systems [20].

4.6. Renewable Energy Policies of Japan

Japan's renewable policy revolution came after the post-tsunami Fukushima Nuclear Power Complex incident in the year 2011. The Japanese government is now earnestly looking forward to establishing as many renewable avenues as possible to mitigate the use of renewable and fossil-fuel based energy systems to ensure energy security and environmental protection. Japan has also become one of the leading contributors to the research and development and integration of ICT with the existing and renewable based grids. Currently, most of the focus is in the direction of hydro based projects and also Boron and Hydrogen fuel cells-based systems [20].

4.7. Renewable Energy Policies of the USA

Despite being one of the largest contributors to RETs, the US still does not have an effective policy regarding renewable energy systems, mainly due to impediments in policy making and capitalist mindset. Currently, both renewable and non-renewable energy systems have been offered similar tax and deployment incentives which has not allowed blooming of renewables to a great extent [20].

5. Conclusion

Renewable Energy has become an unavoidable need for the sustainable future. These systems are important not only for energy security and self-sufficiency but also for the preservation of the global natural environment. In order to ensure deeper and timely penetration of RES as an essential source of energy at a global scale, there is an urgent need to make global and regional policy-makers realize that there is a need to make effective policies to promote the use of RESs in an economical and attractive way. An efficient, stable and reliable energy distribution system calls for increased use of ICT based management, control and storage systems. Policies to mitigate the use of traditional fossil-fuels as a source of energy should also be formulated on priority by imposing penalties on the power-producers heavily relying on fossil-fuels. Policies to promote R&D for more reliable and efficient RESs must be formulated as it would guarantee a long-lasting, clean and stable energy source. The policy makers and relevant regulating authorities must be made independent and fully responsible for achieving RE growth targets. This target-oriented approach would help them work with dedication and independence to achieve the stipulated goals of achieving massive growth in RES based generation capacity and self-sufficiency. Such pro-active and effective policy making bodies could make a big difference regarding the spread and adoption of RES locally and globally.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Music influences a person on a neurological level. It impacts the temperament and conduct of a person [1]. It influences the function of the brain and human actions including, relieving stress, depression symptoms, as well as enhancing cognitive and motor functions, spatial-temporal learning and neurogenesis which is the capacity of the brain to create neurons [2]. The lyrics of the song play a major role in increasing this influence of music. For instance, songs with 'prosocial' lyrics may make you more empathic and could lead to long-term changes in attitudes and behavior for the better [3,4].

The problem arises in creating the lyrics of these songs. Artists face the problem of deciding topics to create songs on. The problem is further intensified in selecting the most unique, catch words which if added, could create more powerful lyrics for the songs. These powerful lyrics with the right combination of other music features (liveliness, loudness, speech ness, etc.) could create hit songs in the future.

A solution for this problem was proposed by this study which analyzed the lyrics of the billboard songs over the past 10 years. This study was carried out using the 'Hot 100 collection of Billboard songs' dataset from 'data world'. This dataset contains every weekly Hot 100 singles chart from 2009-2019. Each row of data represents a song and the corresponding position on that week's chart.

The lyrics of the songs in the dataset were analyzed to find the most important words in each song using TF-IDF technique of natural language processing.

The use of the different type of important words in songs captured the preference of users, being motivated to listen to the most popular hits of the time. This important word's analysis provided a deeper understanding on the choice of words used by the most popular artists.

Furthermore, a unique bag of evergreen words used in song's lyrics from this dataset for the period of 2009-2019 was also found. The lyrics made using these evergreen words could be used to create more powerful songs in the future. This hypothesis of including evergreen words in the song lyrics to create more powerful songs was validated by most desired genre 'Dance Pop' most beloved artist Katy Perry's songs lyrics.

2. Other Works

A study by Petrie et al., [5] used LIWC (Lindquist Inquiry Word Count) for the analysis of the famous band's Beatles songs lyrics during its lifetime. It was observed that the effects of their lyrics were reduced in terms of emotions, focus, orientation, cognitive complexity. They became less social over time.

Another research named 'The Bob Dylan Encyclopedia, New York, Continuum', [6] was focused on repeated use of 'Ecstacy' in rap lyrics and it's changing reference over the decade from 1996. The study found that the increase in the number of raps was directly proportionate to the increasing use of 'Ecstacy' among secondary school teenagers. It also considered changes in songs ciphered with positive, ambiguous, or negative messages about the use of this drug.

In a study Ronce [7], discussed the evolution of Dylan's song lyrics throughout 50 years of his career as an artist by using quantitative and qualitative analyses methods. It analyzed change in his word selection and theme selection explored in his songs in terms of qualitative and quantitative changes respectively. It was found that lyrics of the most popular songs could capture sociocultural changes across generations. A word-count method was used to compute the percentages of words belonging to various psychosocial categories including positive emotion, religion, social processes. Sociocultural studies captured word selection possibly indicating (Dylan's generation) generational changes in American culture. Further examination produced the correlations between the year of album release and the 10 LIWC categories of words. The correlations provided the fact that years were directly proportionate to the type of words used over time.

In another research conducted by Napier et al., [8], the songs were grouped together by year in which they were produced and their tone scores were averaged for each year. The standard deviation and standard error were calculated for every averaged tone of every year. This was followed by conducting two tests to spot a linear relation between features, year and specific tone from the lyrics of the respective year. It was used to examine the relationship between two continuous variables. Furthermore, the linear dependency between these variables was validated using linear regression technique. This research concluded that some of Pearson's correlation coefficients (anger, disgust, fear, and conscientiousness) were considered of having strong positive correlations throughout. This analysis indicated the tonal change of popular music lyrics through generations. Over the years, an increment in some sentiments such as anger, disgust, fear, sadness, tentativeness and conscientiousness and decrement in sentiments such as joy, analytics, confidence and openness was found.

3. Methodology

The Billboard Dataset was used in carrying out the analysis of songs. An additional column of lyrics was added in the dataset to analyze the lyrics of the respective songs. Lyrics were extracted using 'Lyrics Extractor' API in python. The term frequency-inverse document frequency (TF-IDF) scores for the respective lyrics were calculated to find the important words in each song. This word feature selection was based upon selecting words having TF-IDF score greater than the minimum threshold of 0.2. The important words thus found were then visualized using the frequency distribution plot for the respective year in the dataset. This frequency distribution plot was also used in finding the evergreen important words from 2009-2019. These words if added in songs could be used to create more powerful lyrics, thereby helping in creating more hit songs in the future.

This hypothesis was validated by the most desired genre's most beloved artist. A bar chart was visualized to find the most desired genre, i.e., 'Dance Pop'. The most beloved artist, i.e., 'Katy Perry' according to the dataset was found using Pie chart visualized over the top 10 artists of the Dance Pop genre. The lyrics of the songs created by Katy Perry were visualized using word cloud. This word cloud gave an overview of words used in her songs. These words thus found were validated using the bag of unique evergreen words to prove this hypothesis.

Furthermore, a cluster map was plotted to find the correlation between different music features.

3.1. Lyrics extraction

The initial dataset was extended with an additional 'lyrics' parameter. The lyrics were required to perform sentiment analysis on songs. The lyrics for every song were fetched, extracted and added within the dataset using Python's lyrics-extractor Library. This library extracts songs from numerous websites [9]. The library needs the title of the song for the extraction of the lyrics.genius.com web site [10] was used for extracting Billboard Top One Hundred Songs Collection. genius.com has the widest variety of song collections to look from. The GENIUS API (Application Performance Interface) was used for the aim of extracting songs from genius.com.

Genius API follows a collection of protocols, routines and tools for extraction of songs from genius.com web site. Lyrics extractor initially needed a Google API key and an Engine Id of Google custom Search JSON API. An API key is a distinctive selector that is used to validate a user, developer or calling application to an API. The custom search JSON API allows you to build websites and programs to fetch and show search results from Google custom search procedurally, it requires a unique authenticator key to initiate the search process (lyrics extraction). This API was integrated with GENIUS API as mentioned above to extract the lyrics of the specified song from the lyrics-extractor library.

3.2. Computation of term frequency-inverse document frequency scores of vocabulary of song

The term frequency inverse document frequency scores of word features of respective lyrics for each song were computed. This was followed by important word feature selection in the Dataset.

TF-IDF is defined as a statistical technique which evaluates the significance of the respective word in a document within a collection of documents (Dataset). Its applications include in the field of machine learning (Natural Language Processing) and automated text analysis for the computation of scores of word features [11].

It is computed by the multiplication of two metrics namely term frequency and inverse document frequency. The term frequency (TF) of a word in a document is the frequency of a particular word in the document.

The inverse document frequency (IDF) of the word across a set of documents is used to find the respective word's significance in the entire corpus (Dataset). It's closeness to zero is directly proportionate to the frequency of the word in the corpus. It is computed as the logarithmic value of the quotient of the total number of documents and the frequency of documents containing the word by taking the total number of documents.

This produces the TF-IDF score of a word in a document. The score is directly proportionate to the importance of the word in the particular document.

3.3. Extraction of most important features based on term frequency inverse document frequency scores (TF-IDF)

The sum of the TF-IDF score of each word was computed across the 'Lyrics' feature to find the relevance of the word in the dataset. The sum of the scores of word features possessing scores greater than the threshold of 0.2 were selected as the important word features for the respective lyrics of the song in the dataset. The value of 0.2 is conventionally taken as the threshold for selecting important word features based on the TF-IDF scores in the document dataset.

The TF-IDF features thus obtained was used for carrying out the analysis of song lyrics keywords used in lyrics of the songs present in the dataset.

4. Experimental setup

4.1. About dataset

This study uses the 'Hot 100 collection of Billboard songs' dataset from 'data.world' [12]. It contains 650 records divided based on 14 attributes. Some of the attributes included in the dataset are liveness, acoustics ness, energy, loudness, valence, song title, song artists, etc. Additional features lyrics, TF-IDF features were added to complement the analysis of songs present in the dataset.

The lyrics for each song were extracted using lyrics extractor API in python.an API is defined a set of functions and procedures which allows the creation of applications capable of accessing the features or data of an operating system, application, or other services.

Lyrics extraction was followed by finding the most important word of each song in the dataset for broadening analysis on the importance of words in creating lyrics of a song.

4.2. Data preprocessing

The lyrics found using lyrics extractor API were preprocessed for normalization of the 'Lyric' column. The lyrics were normalized using 're', 'nltk' (natural language toolkit) library in python.

Initially only words were selected using the 're' module of python, followed by normalizing the words into lowercase. The words were then converted into their base forms using stemming. Stemming is the process of decreasing a word to its word stem by affixing to suffixes and prefixes or to the roots of words known as a lemma.

This led to the stop words removal from dataset using natural language toolkit library in python. Useless words (data) are referred to as stop words in the dataset.

The corpus of lyrics thus obtained was used for calculating term frequency-inverse document frequency. The sum of the TF-IDF scores was used for word feature selection, selecting the important words in the lyrics in the dataset. The words features having sum greater than threshold value (0.2) computed across the dataset were selected to be used for further analysis and visualization processes.

5. Visualization and analysis of features

The word features obtained after the TF-IDF feature selection technique was used for finding the top 50 words used in songs for every year from 2009-2019. These word features were visualized as frequency distribution plots to support their importance in the dataset. Further analysis led to the discovery of the evergreen words in the past 10 years and the correlation of the features of the dataset using cluster maps and pair plots. Finally, our results for the evergreen words were validated by analyzing the features used in the songs of the most popular pop-dance category's artist Katy Perry.

5.1. Frequency distribution plot

The most important word features (top 50 words) used in lyrics of songs for each year from 2009-2019 were visualized using 'nltk' and 'matplotlib' library in python. For the year 2010, the most important words were 'like', 'wanna', 'go', 'yeah', 'oh', 'love' etc. This was followed by 'girl', 'run', 'yeah', 'baby' etc in 2011 and 'love', 'feeling', 'tonight', 'like', 'talk' in 2019 (Figure 3) and so on. It was also found how the lyrics have changed in the short span of 10 years, as the result of the change in listening preferences of billboard chart listeners.

Analysis on genres of songs and their respective artists revealed that the most popular genre according to the frequency of songs was found to be 'Dance Pop' (Figure 4).

Further it was found that the most beloved artist among the top ten artists was found to be Katy Perry producing 12.8 % of the total billboard songs (Figure 5). On analysis of Katy Perry's songs, it was found that maximum songs produced belonged to the 'Dance Pop' genre.

5.2. Histograms

The histograms of the most popular 'Dance Pop' genre artist Katy Perry songs features were visualized to find the range of frequencies of features like valence, acoustic ness, liveliness, etc.

Katy Perry's songs revealed frequency range of (110-130), (80-85), (60-70), (80-85), (-5.5 - -4.5), (0-2), (4-11) of beats per minute, energy, dance, decibels, acoustic ness, speech ness features respectively (Figure 6).

5.3. Cluster map

The features of the dataset excluding 'TF-IDF-features' (important words) attribute like pop, acoustic ness, liveliness, etc, were explored using cluster map (Figure 7) to gather insights on the correlation or dependence of different features of songs. The cluster map was created 'seaborn' Library in Python.

A cluster map is created using hierarchical clustering techniques to find the amount of similarity or correlation between two features. It was also found that the features energy and loudness (dB) are highly correlated with a positive score of \(+0.54.\) This correlation concluded the fact that people doing workouts listen to loud music to achieve efficiency in their workout.

Acoustics (the characteristic of a space for determination of transmission of sound through it) are not correlated with energy, showcased with a value of \(-0.56\) on the correlation matrix. Liveliness or the probability of live audience listening to the song is not at all correlated with the duration of the song \(+0.11.\) This means that liveliness or the moment a song tries to create does not depend on the length (duration) of the song. It can be felt within the song, during the entire song or not at all in the song.

5.4. Word cloud

A word cloud was made from the most important word features in the 'Dance Pop' star Katy Perry's song's lyrics.

Word cloud also known as a text cloud or a tag cloud is based on simple technique, i.e., the boldness and the size of the word is directly proportionate to the frequency of the word in the specific document. They are generally used for analyzing customer feedback/review and identifying new SEO (Search Engine Optimization) song lyrics keywords to target. The word cloud was created using the 'wordcloud' and 'pandas' library and was visualized using 'matplotlib' library in python. The 'wordcloud' is a library licensed by MIT. It also includes 'DroidSansMono.ttf' apache licensed, a true style font developed by Google [13].

The word cloud generated from Katy Perry's, most beloved artist according to billboard (2009-2019) dataset was visualized to give overall view of the song lyrics keywords used in her songs lyrics as well as for validating the most frequent terms of the word cloud with the most frequent terms from the evergreen words distribution plot. It was found that words like 'love', 'night', 'friday' (also appearing in the top 50 evergreen words) used in Katy Perry's songs were also the most catchy, loveable and desired words for the listeners during the period of 2009-2019 (Figure 8).

These words found in her songs motivated the audience to listen more to these songs. This also led to an increase in the audience for these 'Dance Pop' songs, thereby increasing the number of hits of her songs.

5.5. Sentiment analysis

The song lyrics of Katy Perry's songs were further analyzed on the basis of sentiment, showcased in her songs.

Sentiment analysis is deciphering, analyzing and classifying emotions (like happy, sad, neutral) within textual data using analytical techniques. In methodological terms, sentiment analysis detects polarity within the text, documents, comments etc. It assists organizations in distinguishing an individual's sentiments towards products or services like garments, music etc.

Understanding the emotions through people's point of view is vital as people will communicate their conclusions even more uninhibitedly now in light of digitization. In this way, by analyzing the feedback given by the people, brands will tailor their products and services regarding the people's necessities [14]. The 'vaderSentiment' library was used to find the polarity scores of the songs [15].

The data frame containing 'positive', 'negative', 'neutral', 'compound' scores was plotted as a bar plot using 'pandas' library to visualize the sentiment present in her song's lyrics (Figure 9). This also helped in highlighting the type of sentiment favored by the audience in song's lyrics

6. Results

The popularity of the songs based upon the no of unique evergreen words was plotted and compared with the popularity of songs based on the no of YouTube views (Figure 10). The plot showed that 12 out of 17 songs of the dataset's most beloved artist Katy Perry followed a direct proportionality trend. This trend showed that the number of YouTube views were directly proportional to the no of important words present in the songs. These important words were also present in the unique bag of evergreen words. For instance, a song 'Dark horse' had 83 unique evergreen words in its lyrics and had a 'very popular' likeability (Figure 11). Similarly, another song 'birthday' had only 3 evergreen words and had a 'popular' likeability. This analysis found that the increase in the number of evergreen words made the lyrics more powerful and in turn increased the number of views of the song (popularity).

7. Conclusion

The analysis study found the bag of unique evergreen words using term frequency inverse document frequency and frequency distribution plot. This unique bag of evergreen words contained 2696 words for the songs analyzed from 2009-2019. The top 50 unique evergreen words included 'love', 'like', 'yeah', 'baby', 'let', 'friday', 'night', etc. The hypothesis that lyrics made using these unique evergreen words could be used to create more powerful lyrics in the future was validated using most popular artist Katy Perry's song's lyrics. This analysis also concluded that the no of views (popularity) were positively correlated to the number of words used in the song. The increment in the no of unique evergreen words used in the song also led to an increase in the number of views of the respective songs.

This analysis was also found useful in deciding the genre for creating songs on, which was 'Dance Pop'. The words from the unique bag of evergreen words also could be used in deciding the title on which songs could be created to increase the no of hits of the songs. For example 'last friday night' song by Katy Perry was among the most popular songs. The song lyrics keywords 'friday' and 'night' are also present in bag of unique evergreen words. Similarly a song 'love me like you do' by Ellie Goulding also contains the evergreen words 'like', 'love', 'you' gained large no of hits and was loved by the audience. The title and the category could also be used to capture the sentiment loved by the people (love and like here). The title of the song could be further used to investigate the emotions special to that topic, for example, the 'Friday' could be used to capture the specialty of that day. Similarly, a word 'like' could be used to capture the feeling of being liked or liking something in the songs.

The analysis in future could deepen its understanding of songs using the tonal sequence while analyzing the popularity of the respective song. Furthermore, an increment in the number of songs in the dataset could help in improving the accuracy of this analysis to create more powerful song's lyrics in the future.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction and Preliminaries

A national defense system that has a deterrent power needs to be realized in the development of a defense force capable of protecting and ensuring the fulfillment of the national interests of the Indonesian nation (Ministry of Defence, 2015) [1]. As an archipelagic country with 2/3 of its territorial territory constituting the sea, it is necessary to carry out the development of Navy forces to realize a state defense system at sea [2].

The development of the naval force also needs to be carried out strategically and effectively to achieve the desired deterrence [3]. One of the effective forces to build deterrence in the country's defense strategy at sea is a submarine fleet. Due to the confidentiality, silence, and speed of a submarine that can paralyze a strategic maritime route (maritime chokepoint) and threaten the safety of shipping commercial vessels and naval vessels [4].

The strategic value of the Navy's submarine capabilities is greatly influenced by the ability of its supporting infrastructure to be able to maintain the confidentiality of submarine operations. However, infrastructure development supports the operational resilience of submarines in facing the limitations of the country's capabilities. Based on these conditions, this study aims to identify and analyze the operational readiness of submarines to increase the deterrence of the state defense system at sea. This study uses Interpretive Structural Modeling (ISM) and MIC-MAC diagrams. The ISM method is used to analyze the effect on the criteria. MICMAC diagrams are used as criteria for determining the related variables.

There are several previous texts in supporting research, namely A Maritime Research Concept through Establishing Ship Operational Problem Solution (Shipos) Centre via Information Technologies Integrated With or/Ms [5]. Determinants of Port Performance - Case Study of 4 Main Ports in Indonesia (2005-2015) [6]. Analysis of the Determinants of Micro Enterprises Graduation [7]. Determinants of Indonesian Crude Palm Oil Export: Gravity Model Approach [8]. ISM for analyzing interactions between barriers to just-in-time (JIT) production operations [9]. ISM to an analysis of core industry competencies in Pekalongan City [10]. ISM uses for identification of readiness in implementing Business Intelligence projects [11]. ISM to identify the drivers of travel/tourism growth and build relationships between enablers [12]. ISM to study various aspects and correlations between youth and sustainable rural development [13]. Navy Ability Development Strategy using SWOT Analysis-Interpretative Structural Modeling (ISM) [14].

This research is limited to the operational durability of submarines in Indonesia. This research is expected to provide benefits to both academics and practitioners in the field of submarines and defense strategy, Developing knowledge about the management of the national defense system.

This research consists of several parts. Section 2 describes ship operations during the state defense system, ISM method, MICMAC diagram, research stages, and research subjects & objects. Section 3 describes the influence of variables in submarine operations and the classification of the distribution of these variables. Section 4 describes the conclusions in the study.

2. Material & Methods

2.1. State defense system at sea

State defense aims to safeguard and protect the sovereignty of the country, the territorial integrity of the Republic of Indonesia, and the safety of the entire nation from all forms of threats and disturbances, both from outside and from within the country [15]. However, as a nation that has high-value national resources, vigilance is needed to continue to anticipate all forms of threats that are dynamic and can turn into real threats to ensure protection and fulfillment of national interests and to uphold national sovereignty (Ministry of Defence, 2015) [1].

Seapower is projected to control the use of marine areas, trade, and commerce at sea, as well as for deterrence, resistance at sea, defense diplomacy, and exerting political influence. The use of naval power is generally defined as the trinity of the role of maritime security forces which is manifested in the strength of a country's Navy [16].

The Navy has a unique ability compared to other militaries, namely the ability to produce coercive signals and deterrence to the enemy. Naval power can effectively transmit these signals, and their effectiveness stems from two unique features of the Navy's capabilities. Referring to the nature and geographical conditions of the Indonesian nation, the development and development of the Navy's strength is a strategic policy analysis that needs attention in building a deterrent state defense system [17].

2.2. Submarines as a state defense system

In the context of national defense at sea, the strength of the Navy fleet is the domain in the formation, response, and preparation of all national resources to build a national defense system. The development of security and defense challenges at sea is currently felt to increasingly require the existence of a submarine fleet that is sufficient in quantity, modern technology, and operationally sustainable. The quantity aspect relates to the ideal number of submarines owned by a country. This number is greatly influenced by the size of the sea area owned by a country and the maritime defense strategy carried out by that country (Ministry of Defence, 2015) [1].

The submarine will be strategically connected to the command and control system where all warning and sensor systems will be the basis for providing command. The submarine fleet has certain advantages that surface warships do not have. Operational advantages include; can be used to bring troops into special operations fields; can be used to collect intelligence data as well as underwater surveys related to oil and natural gas mining sources; can be more effective in pursuing enemy warships during sea battles; as well as in terms of weaponry it is considered more accurate in carrying out subsurface and torpedo missile launches for both warships, submarines, and surface targets on land [18].

The strategic value of the submarine fleet can also be proven at an operational level that is not owned by surface warships or land and air warfare vehicles. These operational advantages include its ability to be used as a means of bringing troops into special operations in full confidence; can be used to collect intelligence data as well as underwater surveys related to oil and natural gas mining sources; can be more effective in destroying the strength of the enemy warship fleet during sea battles; as well as in terms of weaponry it is considered more accurate in carrying out subsurface and torpedo missile launches for both warships, submarines, and surface targets on land (Ministry of Defence, 2015) [1].

2.3. Submarine operations

The submarine is a unique water vehicle due to its ability to present strength in conventional and asymmetric warfare. This capability is achieved through its ability to maintain the confidentiality of the submarine's existence, which is related to the technology installed on the water vehicle. Determination of the technology that needs to be installed on a submarine generally considers the operational task to be carried out, the area of operation, the range of transit times, speed, diving depth, passive defense, weapon systems, rescue and rescue, re-support support, maintenance, and average size the crew [19].

The above technological factors can be broadly grouped into ship hull technology (platform), machinery system technology, propulsion system technology, navigation and communication system technology, and weapon systems technology. The sustainability of submarine operations is also determined by the technological aspects of the support equipment in the submarine maintenance and repair system owned by a country's navy. Submarine maintenance and repair support technology involves several main indicators, such as the submarine operation support logistics system, submarine maintenance and repair systems, and submarine operational financing systems [20].

Thus, the existence of supporting equipment technology, which includes a logistics system to support the submarine movement, policies related to the use, maintenance, and repair of submarines, as well as technical aspects which include the use of technology and submarine maintenance and repair facilities can be used as indicators key both in the framework of developing a strategy formula and implementing a sustainable submarine operational strategy [21].

2.4. Interpretive structural modeling (ISM)

Interpretive structural modeling (ISM) is used for ideal planning, which is an effective method because all elements can be processed in a simple matrix [22]. ISM was first proposed by Warfield in 1973. Interpretive structural modeling is a methodology that aims to identify the relationships between certain items, which define a related problem or issue [23], and modeling techniques suitable for analyzing the effect of one variable on another [24].

ISM is well-proven to identify structural relationships among system-specific variables. The basic idea is to use practical experience and expert knowledge to break down a complex system into sub-systems (elements) and build a multilevel structural model [25]. The ISM-based approach is one of the most versatile and powerful techniques that have been used to solve complex multi-factor problems. ISM is interpretive, in that the assessment of the group selected for research determines whether and how the variables are related [26]. ISM is often used to provide a fundamental understanding of a complex situation, as well as to construct a series of actions to solve a problem. There are procedures or stages in using the ISM method, these stages include [25]:

• 1. Parameter identification.

  The elements to be considered for the identification of relationships were obtained through literature surveyors by conducting surveys.

• 2. Development of Structural Self Interaction Matrix (SSIM).

  The development of the interpretive structural model begins with the construction of a structural self-interaction matrix, which shows the direction of contextual relationships among elements. In developing SSIM, the following four symbols have been used to indicate the direction of the relationship between two constraints i and j.

• 3. Matriks Reachability. From the self-interaction matrix (SSIM), relational indicators are converted into binary numbers 0 and 1 to obtain a square matrix, which is called the reachability matrix [27].
  • If (i, j) the value in SSIM is V, (i, j) the value in the reachability matrix will be 1, and (j, i) the value will be 0.
  • If (i, j) the value in SSIM is A, (i, j) the value in the reachability matrix will be 0, and (j, i) the value will be 1.
  • If (i, j) the value in SSIM is X, (i, j) the value in the reachability matrix will be 1, and (j, i) the value will also be 1.
  • If (i, j) the value in SSIM is the value of O, (i, j) in the reachability matrix will be 0, and (j, i) will also be 0.
• 4. Partition level.

  From the reachability matrix, for each parameter, the reachability set and the antecedent set are derived. Variables, which are common in reachability sets and antecedent sets, are allocated to intersection sets. Once the upper-level barrier is identified, it is removed from consideration, and other upper-level barriers are found [25]. This process will continue until all levels of each barrier are found.

• 5. Interpretive structural modeling constructs (ISM).

  From the partitioned parameters and reachability matrix, the structured model is derived, showing the parameters at each level and arrows showing the direction of the relationship.

Table 1. SSIM symbol rules.

Symbol	Relationship Between Element Row (i) and Column (j)
V	There is a contextual relationship between the \(E_i\) element and the
	\(E_j\) element, but it is not the other way around.
A	There is a contextual relationship between the element \(E_j\) and the element \(E_i\), but not vice versa.
X	There is a reciprocal contextual relationship between \(E_i\) elements and \(E_j\) elements.
O	There is no reciprocal contextual relationship between element \(E_i\) and element \(E_j\).

3. MIC-MAC Diagram

MICMAC analysis refers to Matrice d'Impacts Croisés Multiplication Appliquée á un Classement (Hussain, 2011) and involves developing a graph to classify the various enablers based on their driving power and dependence power. MICMAC is also used to check driving power and power dependence so that later it can be identified which elements are the keys to driving the system being analyzed. The variables have been classified into four categories referred to as Autonomous, Linkage, Dependent and Driving/independent.

Table 2. Samples on the reachability matrix.

(i)Enablers	1	2	3	4	5	6	7	8
1	1	1	0	0	0	0	0	0
2	0	1	1	0	0	0	0	0
3	1	1	1	1	1	1	1	0
4	1	1	1	1	1	1	1	0
5	1	1	1	1	1	1	1	0
6	1	1	1	1	1	1	1	0
7	1	1	1	1	1	1	1	0
8	1	1	1	1	1	1	1	1

3.1. Research Subject and Obyek

The object of this research is the operational resilience of the submarine. Submarine operational resilience is defined as the ability of a submarine to be in the area of operation without being noticed by other parties in conditions of diving (diving depth), diving periscope (periscope depth), or sailing on the surface.

Indicators used to collect data from resource persons about the influence of various conditions (the system in the submarine, the mode of sailing during operation, physical and psychological readiness of the crew, availability of operational budgets, and the development of the strategic environment of the Indonesian nation) on the operational durability of the submarine.

3.2. Research subject

Research subjects or respondents are parties who are used as samples in a study. The research subject also discusses the characteristics of the subject used in the study, including an explanation of the population, sample, and sampling technique used. In this study, the sources of research were experts, including 1) Vice Admiral Muhammad Ali; 2) Marx Jefferson; 3) Vice Admiral (Rtr) Rachmad Lubis; 4) Vice Admiral Iwan Isnurwanto; 5) Vice Admiral TNI Tunggul Suropati.

Table 3. SSIM symbol rules.

No.	Expert	Total	Code
1	Commander of First Fleet Command, Vice Admiral Muhammad Ali	1	\(E_{1}\)
2	Head of the Submarine Program of PT PAL Indonesia, Marx Jefferson	1	\(E_{1}\)
3	Head of the KKIP Technology Transfer and Offset Division,	1	\(E_{3}\)
	Vice Admiral (Rtr) Rachmad Lubis
4	Vice Admiral Iwan Isnurwanto	1	\(E_{4}\)
5	Vice Admiral TNI Tunggul Suropati	1	\(E_{5}\)

The object of research is essentially the topic of the problem understudy in the research. The object of research is essentially the topic of the problem understudy in the research. In this study, the object of research is the durability of submarine operations in Indonesia.

3.3. Goal

In this study, the target is the influence of the existing criteria and classify them in the variables contained in submarine operational durability.

4. Results

The data collected from the questionnaires were compiled in the research database used in the subsequent data analysis process. After the data was collected, the research instrument was tested to obtain confidence that each question item was able to accurately and accurately be used as a variable measuring tool.

The submarine operational survival variable is measured (observable variables) through indicators fuel tank capacity; freshwater tank capacity; submarine battery capacity; air regeneration system capacity; sail with surface mode/periscope depth / operational depth; operating area wave conditions; the physical properties of the water column (temperature, salinity, depth); crew physical readiness; the psychic readiness of the crew; Operating budget; Operation pattern; Strategic Environment Development Conditions.

Table 4. Elements of Ketahanlamaan Operasional Kapal Selam.

Code	Criteria
\(C_1\)	Fuel Tank Capacity
\(C_2\)	Fuel Tank Capacity
\(C_3\)	Submarine Battery Capacity
\(C_4\)	Air Regeneration System Capacity
\(C_5\)	Sail With Surface Mode
\(C_6\)	Sail With A Periscope Depth
\(C_7\)	Sail With Operational Depth
\(C_8\)	Operating Area Wave Conditions
\(C_{9}\)	Current conditions of the operating area
\(C_{10}\)	The Physical Properties Of The Water Column (Temperature, Salinity, Depth)
\(C_{11}\)	Crew Physical Readiness
\(C_{12}\)	The Psychic Readiness Of The Crew
\(C_{13}\)	Operating Budget
\(C_{14}\)	Operation Pattern
\(C_{15}\)	Strategic Environment Development Conditions

The second step, application of identified criteria or variables that are defined in pairs. Then with SSIM, paired relationships are developed between the factors that affect the system. The association matrix is evaluated by SSIM and used for transitivity within the ISM. Next, factorization was carried out on the related criteria. Data recapitulation of interconnection between strategies is analyzed in the Structural Self Interaction Matrix (SSIM) by converting numbers into letters that represent the categories of relationships. At this stage, a contextual relationship is made between variable \(i\) and variable \(j\). Next, review the contextual relationship in the form of the SSIM-VAXO Matrix.

Table 5. Results of the structural self interaction matrix (SSIM) analysis.

No	Code	Factorization
		15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
1	\(C_{1}\)	O	A	A	O	O	O	O	O	V	V	V	O	O	O
2	\(C_{2}\)	O	O	V	O	O	V	O	O	V	V	V	O	O
3	\(C_{3}\)	O	V	O	O	O	O	O	O	V	V	V	O
4	\(C_{4}\)	O	O	O	O	V	X	O	O	V	V	V
5	\(C_{5}\)	O	A	O	V	V	A	A	A	X	X
6	\(C_{6}\)	O	O	O	A	A	A	A	X	X
7	\(C_{7}\)	O	A	O	V	V	A	A	A
8	\(C_{8}\)	O	V	O	V	V	X	X
9	\(C_{9}\)	A	V	O	V	V	A
10	\(C_{10}\)	O	V	O	O	O
11	\(C_{11}\)	O	V	A	X
12	\(C_{12}\)	O	A	A
13	\(C_{13}\)	A	X
14	\(C_{14}\)	A
15	\(C_{15}\)

In the second stage, the reachability matrix table is generated by the symbols V, O, A, and X to become binary 1 and 0, with the following conditions:

• If relation \((i, j)\) is denoted as V, then input \((i, j)\) in RM becomes 1, and input \((j, i)\) becomes 0.
• If relation \((i, j)\) is denoted as A, then input \((i, j)\) in RM becomes 0, and input \((j, i)\) becomes 1.
• If the relation \((i, j)\) is denoted as X, then input \((i, j)\) in RM becomes 1, and input \((j, i)\) becomes 1.
• If the relation \((i, j)\) is denoted as O, then input \((i, j)\) in RM becomes 0, and input \((j, i)\) becomes 0.

Table 6. Results of the reachability matrix.

No.	Code	Factorization															DP
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	\(C_1\)	1	0	0	0	1	1	1	0	0	0	0	0	0	0	0	4
2	\(C_2\)	0	1	0	0	1	1	1	0	0	1	0	0	1	0	0	6
3	\(C_3\)	0	0	1	0	1	1	1	0	0	0	0	0	1	0	0	5
4	\(C_4\)	0	0	0	1	1	1	1	0	0	1	1	0	0	0	0	6
5	\(C_5\)	0	0	0	0	1	1	1	0	0	0	1	1	0	0	0	5
6	\(C_6\)	0	0	0	0	1	1	1	1	0	0	0	0	0	0	0	4
7	\(C_7\)	0	0	0	0	1	1	1	0	0	0	1	1	0	0	0	5
8	\(C_8\)	0	0	0	0	1	1	1	1	1	1	1	0	0	1	0	9
9	\(C_9\)	0	0	0	0	1	1	1	1	1	0	1	0	0	1	0	8
10	\(C_{10}\)	0	0	0	1	1	1	1	1	1	1	0	0	0	1	0	8
11	\(C_{11}\)	0	0	0	0	0	1	0	0	0	1	1	1	0	1	0	5
12	\(C_{12}\)	0	0	0	0	0	1	0	0	0	1	1	1	0	0	0	4
13	\(C_{13}\)	1	0	0	0	0	0	0	0	0	0	1	1	1	1	0	5
14	\(C_{14}\)	1	0	0	0	1	1	0	0	0	0	0	1	1	1	0	6
15	\(C_{15}\)	0	0	0	0	0	0	0	0	1	0	0	0	1	1	1	4
Dependence		3	1	1	2	11	13	10	4	4	6	8	8	5	7	1

4.1. MICMAC Analysis

The next step is to arrange a hierarchy of each sub-element in the element being studied and classify it into four sectors, whether these sub-elements are included in the Autonomous, Dependent, Linkage or Independent sectors, namely:

• 1. Quadrant-I: weak driver-weak dependent variables (Autonomous), which means that the sub-elements that enter this sector are generally not related to the system and may have a little relationship even though the relationship can be strong.
• 2. Quadrant-II: weak driver-strongly dependent variables (dependent) which means that the sub-elements in this sector are dependent.
• 3. Quadrant-III: strong driver-strongly dependent variables (linkage), which means that the sub-elements that enter this sector must be studied carefully because the relationship between the sub-elements is unstable.
• 4. Quadrant-IV: Strong driver-weak dependent variables (Independent), which means that the sub-elements that enter this sector are the remaining parts of the system which are called independent variables.

Table 7. Driving power & dependence.

	\textbf{\(C_1\)}	\textbf{\(C_4\)}	\textbf{\(C_3\)}	\textbf{\(C_{4}\)}	\textbf{\(C_5\)}	\textbf{\(C_6\)}	\textbf{\(C_7\)}	\textbf{\(C_8\)}	\textbf{\(C_9\)}	\textbf{\(C_{10}\)}	\textbf{\(C_{11}\)}	\textbf{\(C_{12}\)}	\textbf{\(C_{13}\)}	\textbf{\(C_{14}\)}	\textbf{\(C_{15}\)}
\textbf{X}	3	1	1	2	11	13	10	4	4	6	8	8	5	7	1
\textbf{Y}	4	6	5	6	5	4	5	9	8	8	5	4	5	6	4

The result of the MICMAC diagram analysis classifies the elements into four sectors in a two-dimensional graph with the x (dependence) and y (powder driver) axes.

Based on Figure 4, obtained several classifications of submarine operational durability elements. These elements are divided into four classifications, namely:

• 1. Quadrant-I (Autonomous) consists of six elements, namely: a) Fuel tank capacity \((C_1)\); b) Freshwater tank capacity \((C_2)\); c) Submarine battery capacity \((C_3)\); d) Air regeneration system capacity \((C_4)\); e) Operating budget \((C_{13})\); f) Strategic Environment Development Condition \((C_{15})\).
• 2. Quadrant-II (Dependent) consists of six elements, namely: a) Sailing in surface mode \((C_5)\); b) Sailing with periscope depth \((C_6)\); c) Sailing with operational depth \((C_7)\); d) Crew Physical Readiness \((C_{11})\); e) The Psychic Readiness Of The Crew \((C_{12})\); f) Operation pattern \((C_{14})\).
• 3. Quadrant-III (Linkage) there are no elements.
• 4. Quadrant-IV (Independent) consists of three elements, namely: a) Operating area wave conditions \((C_8)\); b) Current conditions in the operating area \((C_9)\); c) Physical properties of the water column (temperature, salinity, depth) \((C_{10})\).

5. Conclusion

This study aims to identify and analyze the operational readiness of submarines to increase the deterrence of the state defense system at sea. Based on the research results, there are several classifications of submarine operational durability elements. Based on the results of the ISM and MICMAC diagram analysis, it was found that fifteen elements were divided into 4 (four) quadrants, likely Quadrant I (Autonomous) consists of six elements, namely: a) Fuel tank capacity \((C_1)\); b) Freshwater tank capacity \((C_2)\); c) Submarine battery capacity \((C_3)\); d) Air regeneration system capacity \((C_4)\); e) Operating budget \((C_{13})\); f) Strategic Environment Development Condition \((C_{15})\). Quadrant II (Dependent) consists of six elements, namely: a) Sailing in surface mode \((C_5)\); b) Sailing with periscope depth \((C_6)\); c) Sailing with operational depth \((C_7)\); d) Crew Physical Readiness \((C_{11})\); e) The Psychic Readiness Of The Crew \((C_{12})\); f) Operation pattern \((C_{14})\). Quadrant IV (Independent) consists of three elements, namely: a) Operating area wave conditions \((C_8)\); b) Current conditions in the operating area \((C_9)\); c) Physical properties of the water column (temperature, salinity, depth) \((C_{10})\).

Future Work

There are advised in research that can be used for future research, among others:

• As advised to carry out an analysis of operational durability using the Structural Equation Model (SEM) method.
• As advised, an evaluation analysis of current submarine operational policies can be carried out.
• As further research, a submarine operational strategy concept can be made following the development of the industrial revolution 4.0 and naval technology 2030.

Acknowledgments

This Research is supported by Indonesia Defense University. We also thank all parties who support this research.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction and Preliminaries

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra. Other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

The concept of a fuzzy set was introduced by Zadeh [1], and it is now a rigorous area of research with manifold applications ranging from engineering and computer science to medical diagnosis and social behavior studies. Conorms are operations which generalize the logical conjunction and logical disjunction to fuzzy logic. They are a natural interpretation of the conjunction and disjunction in the semantics of mathematical fuzzy logics and they are used to combine criteria in multi-criteria decision making. Yuan and Lee [2] defined the fuzzy subgroup and fuzzy subring based on the theory of falling shadows. Also Solairaju and Nagarajan [3] introduced the notion of \(Q\)- fuzzy groups. The author by using norms, investigated some properties of fuzzy algebraic structures [4,5,6].

In [7] the author introduced the notion of anti \(Q-\)fuzzy subgroups of \(G\) with respect to t-conorm \(C\) and study their important properties. In this work, we introduce strongest relation with respect anti \(Q\)-fuzzy subgroups of \(G\) with respect to \(t\)-conorm \(C\) and obtain some properties of them. Next, we define the middle coset of anti \(Q\)-fuzzy subgroups of \(G\) with respect to \(t\)-conorm \(C\) and investigate some results about them. Finally, we define new group under new operations of them and we prove isomorphism between them.

2. Preliminaries

In this section, we recall some of the fundamental concepts and definition, which are necessary for this paper. For more details we refer readers to [8,9,10,11].

Definition 1. A group is a non-empty set \( G\) on which there is a binary operation \((a,b) \to ab\) such that

1. if \( a \) and \( b\) belong to \( G \) then \( ab\) is also in \( G\) (closure),
2. \(a(bc) = (ab)c\) for all \(a,b,c \in G\) (associativity),
3. there is an element \( e \in G\) such that \(ae = ea = a\) for all \(a \in G\) (identity),
4. if \(a \in G,\) then there is an element \(a^{-1} \in G\) such that \(aa^{-1} = a^{-1}a =e\) (inverse).

One can easily check that this implies the unicity of the identity and of the inverse. A group \( G \) is called abelian if the binary operation is commutative, i.e., \(ab = ba\) for all \(a,b \in G.\)

Remark 1. There are two standard notations for the binary group operation, either the additive notation, that is \((a,b) \to a + b\) in which case the identity is denoted by \( 0\), or the multiplicative notation, that is \((a,b) \to ab\) for which the identity is denoted by \( e. \)

Definition 2. Let \( (G,.) , (H,.) \) be any two groups. The function \( f:G \to H\) is called a homomorphism (anti-homomorphism) if \( f(xy)=f(x)f(y) (f(xy)=f(y)f(x)),\) for all \( x,y\in G.\)

Definition 3. Let \(G\) be an arbitrary group with a multiplicative binary operation and identity \(e\). A fuzzy subset of \(G\), we mean a function from \(G\) into \( [0,1]\). The set of all fuzzy subsets of \(G\) is called the \( [0,1]\)-power set of \(G\) and is denoted \( [0,1]^G.\)

Definition 4. A \(t\)-conorm \(C\) is a function \(C : [0,1]\times [0,1] \to [0,1]\) having the following four properties:

• (C1) \(C(x,0)=x\),
• (C2) \(C(x,y)\leq C(x,z)\) if \(y\leq z\),
• (C3) \(C(x,y)= C(y,x)\),
• (C4) \( C(x,C(y,z))=C(C(x,y),z)\), for all \(x,y,z \in [0,1].\)

Example 1. Following are some examples of \(t\)-conorms:

1. Standard union \(t\)-conorm \(C_m(x,y) = \max \{ x,y \}.\)
2. Bounded sum \(t\)-conorm \(C_b(x,y) =\min\{1, x+y \}.\)
3. Algebraic sum \(t\)-conorm \(C_p(x, y) = x+y-xy. \)
4. Drastic \(T\)-conorm \begin{equation*} C_{D}(x,y) = \left\{ \begin{array}{rl} y &\text{if } x=0\\ x &\text{if } y=0\\ 1 &\text{otherwise,} \\ \end{array} \right. \end{equation*} dual to the drastic \(T\)-norm.
5. Nilpotent maximum \(T\)-conorm, dual to the nilpotent minimum \(T\)-norm: \begin{equation*} C_{nM}(x,y) = \left\{ \begin{array}{rl} \max \lbrace x , y \rbrace &\text{if } x+y < 1\\ 1 &\text{otherwise. } \\ \end{array} \right. \end{equation*}
6. Einstein sum (compare the velocity-addition formula under special relativity) \(C_{H_{2}}(x,y)=\dfrac{x+y}{1+xy}\) is a dual to one of the Hamacher \(t\)-norms. Note that all \(t\)-conorms are bounded by the maximum and the drastic t-conorm: \( C_{\max} ( x , y ) \leq C(x,y) \leq C_{D}(x,y)\) for any \(t\)-conorm \(C\) and all \( x,y \in [0,1].\)

Recall that \(t\)-conorm \(C\) is idempotent if for all \(x \in [0,1]\), we have that \(C(x,x)=x.\)

Lemma 1. Let \(C\) be a \(t\)-conorm, then \[C(C(x,y),C(w,z))= C(C(x,w),C(y,z)),\] for all \(x,y,w,z\in [0,1].\)

Definition 5. Let \(( G, . )\) be a group and \(Q\) be a non empty set. \( \mu \in [0,1]^{G \times Q} \) is said to be an anti \(Q\)-fuzzy subgroup of \(G\) with respect to \(t\)-conorm \(C\) if the following conditions are satisfied:

1. \( \mu(xy,q) \leq C(\mu(x,q),\mu(y,q)),\)
2. \( \mu(x^{-1},q) \leq \mu(x,q),\) for all \( x,y \in G \) and \( q\in Q. \)

Throughout this paper the set of all anti \(Q\)-fuzzy subgroups of \(G\) with respect to \(t\)-conorm \(C\) will be denoted by \(AQFSC(G).\)

Lemma 2. Let \(\mu \in AQFSC(G)\) and \(C\) be idempotent \(t\)-conorm, then \(\mu(e_{G},q) \leq \mu(x,q)\) for all \(x\in G\) and \( q\in Q.\)

Proposition 1. Let \(C\) be idempotent \(t\)-conorm, then \(\mu \in AQFSC(G)\) if and only if \[\mu(xy^{-1},q) \leq C(\mu(x,q) , \mu(y,q))\] for all \(x,y\in G\) and \(q\in Q.\)

Definition 6. We say that \(\mu\in AQFSC(G)\) is a normal if \(\mu(xyx^{-1},q) = \mu (y,q)\) for all \(x,y\in G\) and \( q\in Q. \) We denote by \(NAQFSC(G)\) the set of all normal anti \(Q\)-fuzzy subgroups of \(G\) with respect to \(t\)-conorm \(C.\)

Definition 7. Let \( (G,.) , (H,.) \) be any two groups such that \(\mu \in AQFSC(G)\) and \(\nu \in AQFSC(H).\) The product of \( \mu \) and \( \nu\), denoted by \( \mu \times \nu \in [0,1]^{(G \times H) \times Q}\) , is defined as \[ (\mu \times \nu) ((x,y),q)= C(\mu(x,q) , \nu(y,q))\] for all \( x\in G, y\in H,q\in Q. \)

Throughout this paper, \(H\) denotes an arbitrary group with identity element \(e_{H}.\)

Proposition 2. If \(\mu \in AQFSC(G)\) and \(\nu \in AQFSC(H),\) then \( \mu \times \nu \in AQFSC(G \times H).\)

3. strongest relation, cosets and middle cosets of \(QFST(G)\)

Definition 8. Let \( \mu \in [0,1]^{G \times Q}\) and \( \nu \in [0,1]^{(G \times G) \times Q}.\) We say that \( \nu \) is the strongest relation of \( G\) with respect to \( \mu\) if \[ \nu((x,y),q)=C(\mu(x,q),\mu(y,q)).\]

Proposition 3. Let \( \mu \in [0,1]^{G \times Q}\) and \( \nu \) be the strongest relation of \( G\) with respect to \( \mu.\) Then \( \mu \in AQFSC(G)\) if and only if \( \nu \in AQFSC(G \times G).\)

Proof. Let \( \mu \in AQFSC(G),\) then,

1. for all \((x_{1},x_{2}), (y_{1},y_{2}) \in G \times G\) and \( q \in Q \) we have \begin{align*}\nu((x_{1},x_{2})(y_{1},y_{2}),q)&=C(\mu(x_{1},x_{2}),q),\mu(y_{1},y_{2}),q))\\ &\leq C(C(\mu(x_{1},q),\mu(x_{2},q)),C(\mu(y_{1},q),\mu(y_{2},q)))\\ &=C(C(\mu(x_{1},q),\mu(y_{1},q)),C(\mu(x_{2},q),\mu(y_{2},q)))\\ &=C(\nu((x_{1},y_{1}),q),\nu((x_{2},y_{2}),q))\end{align*} and then \[\nu((x_{1},x_{2})(y_{1},y_{2}),q) \leq C(\nu((x_{1},y_{1}),q),\nu((x_{2},y_{2}),q)).\]
2. For all \((x,y) \in G \times G\) and \( q \in Q \) we obtain that \begin{align*} \nu((x,y)^{-1},q)&=\nu((x^{-1},y^{-1}),q)\\ &=C(\mu(x^{-1},q),\mu(y^{-1},q))\\ &\leq C(\mu(x,q),\mu(y,q))\\& =\nu((x,y),q).\end{align*} Then \( \nu \in AQFSC(G \times G).\)

Conversely, assume that \( \nu \in AQFSC(G \times G).\)

1. Let \( x_{1}, x_{2}, y_{1}, y_{2} \in G \) with \(x_{2}=y_{2}=e_{G}\) and \(q \in Q.\) From Lemma 2 we get that \( \mu(e,q) \leq \mu(x_{1}y_{1},q) \) and so \begin{align*}\mu(x_{1}y_{1},q) &=C(\mu(x_{1}y_{1},q),\mu(e_{G},q))\\ &=C(\mu(x_{1}y_{1},q),\mu(x_{2}y_{2},q))\\ &=\nu((x_{1}y_{1},x_{2}y_{2}),q)\\ &=\nu((x_{1},x_{2})(y_{1},y_{2}),q)\end{align*}\begin{align*} &\leq C(\nu((x_{1},x_{2}),q),\nu((y_{1},y_{2}),q))\\ &=C(C(\mu(x_{1},q),\mu(x_{2},q)),C(\mu(y_{1},q),\mu(y_{2},q)))\\ &=C(C(\mu(x_{1},q),\mu(e,q)),C(\mu(y_{1},q),\mu(e,q))) \\ &\leq C(C(\mu(x_{1},q),\mu(x_{1},q)),C(\mu(y_{1},q),\mu(y_{1},q)))\\ &=C(\mu(x_{1},q),\mu(y_{1},q)))\end{align*} and thus \[\mu(x_{1}y_{1},q) \leq C(\mu(x_{1},q),\mu(y_{1},q)).\]
2. Let \(x,y \in G\) with \( y=e_{G} \) and \( q \in Q. \) Then by Lemma 2 we obtain \begin{align*} \mu(x^{-1},q)&=C(\mu(x^{-1},q),\mu(e_{G},q)) \\ &\leq C(\mu(x^{-1},q),\mu(y^{-1},q))\\ &=\nu((x^{-1},y^{-1}),q)\\ &=\nu((x,y)^{-1},q) \leq \nu((x,y),q)\\ &=C(\mu(x,q),\mu(y,q))\\ &=C(\mu(x,q),\mu(e,q))\\ & \leq C(\mu(x,q),\mu(x,q))\\ &=\mu(x,q)\end{align*} and then \[\mu(x^{-1},q) \leq \mu(x,q).\] Therefore \( \mu \in AQFSC(G).\)

Definition 9. Let \( \mu \in AQFSC(G)\), then middle coset \( a\mu b: G \times Q \to [0,1]\) is defined by \[(a \mu b)(x,q)=\mu(a^{-1}xb^{-1},q)\] for all \( x \in G, q \in Q \) and \( a, b \in G. \)

Proposition 4. Let \( \mu \in AQFSC(G),\) then \( a \mu a^{-1} \in AQFSC(G)\) for any \( a \in G. \)

Proof. Let \( a, x, y \in G \) and \( q \in Q, \) then

1. \((a \mu a^{-1})(xy,q)=\mu(a^{-1}xya,q)=\mu(a^{-1}xaa^{-1}ya,q) \leq C(\mu(a^{-1}xa,q),\mu(a^{-1}ya,q))\\=C((a \mu a^{-1})(x,q),(a \mu a^{-1})(y,q)),\)
2. \((a \mu a^{-1})(x^{-1},q)=\mu(a^{-1}x^{-1}a,q)=\mu((a^{-1}xa)^{-1},q)\leq \mu(a^{-1}xa,q)=(a \mu a^{-1})(x,q).\)

Thus \( a \mu a^{-1} \in AQFSC(G)\) for any \( a \in G. \)

Definition 10. Let \( \mu \in AQFSC(G)\), then coset \( a\mu : G \times Q \to [0,1]\) is defined by \((a \mu)(x,q)=\mu(a^{-1}x,q)\) for all \( x \in G, q \in Q \) and \( a \in G. \)

Proposition 5. Let \( \mu \in AQFSC(G)\) and \(C\) be idempotent \(t\)-conorm, then \( x\mu=y\mu \) if and only if \( \mu(x^{-1}y,q)=\mu(y^{-1}x,q)=\mu(e_{G},q)\) for all \( x,y \in G \) and \( q \in Q. \)

Proof. Let \( x,y, g \in G.\) If \( x\mu=y\mu,\) then \(x \mu(x,q)= y \mu(x,q)\) and \( \mu(x^{-1}x,q)=\mu(y^{-1}x,q) \) so \( \mu(e_{G},q)= \mu(y^{-1}x,q).\)

Also as \( x\mu=y\mu\) so \(x \mu(y,q)= y \mu(y,q)\) and \( \mu(x^{-1}y,q)=\mu(y^{-1}y,q) \), which implies that \( \mu(x^{-1}y,q)=\mu(e_{G},q).\) Therefore we obtain that \( \mu(x^{-1}y,q)=\mu(y^{-1}x,q)=\mu(e_{G},q). \)

Conversely, let \( \mu(x^{-1}y,q)=\mu(y^{-1}x,q)=\mu(e_{G},q). \) Then

\begin{align*} x\mu(g,q)&=\mu(x^{-1}g,q)\\ &=\mu(x^{-1}yy^{-1}g,q) \\ &\leq C(\mu(x^{-1}y,q),\mu(y^{-1}g,q))\\ &=C(\mu(e_{G},q),\mu(y^{-1}g,q)) \\ &\leq C(\mu(y^{-1}g,q),\mu(y^{-1}g,q))\\ &=\mu(y^{-1}g,q)=y\mu(g,q)\\ &=\mu(y^{-1}g,q)\\ &=\mu(y^{-1}xx^{-1}g,q)\\ &\leq C(\mu(y^{-1}x,q),\mu(x^{-1}g,q))\\ &=C(\mu(e_{G},q),\mu(x^{-1}g,q))\\ & \leq C(\mu(x^{-1}g,q),\mu(x^{-1}g,q))\\ &=\mu(x^{-1}g,q)=x\mu(g,q)\end{align*} and \( x\mu(g,q)=y\mu(g,q) \), which implies that \( x\mu=y\mu.\)

Proposition 6. Let \( \mu \in AQFSC(G)\) and \(C\) be idempotent \(t\)-conorm. If \( x\mu=y\mu,\) then \( \mu(x,q)=\mu(y,q) \) for all \( x,y \in G \) and \( q \in Q. \)

Proof. Since \( x\mu=y\mu,\) so from Proposition 5 we have \( \mu(x^{-1}y,q)=\mu(y^{-1}x,q)=\mu(e_{G},q)\) for all \( x,y \in G \) and \( q \in Q. \) Now \begin{align*} \mu(x,q)&= \mu(yy^{-1}x,q)\\ &\leq C( \mu(y,q),\mu(y^{-1}x,q))\\ &=C( \mu(y,q),\mu(e_{G},q)) \\ &\leq C( \mu(y,q),\mu(y,q))\\ &=\mu(y,q)\\ &=\mu(xx^{-1}y,q)\\ & \leq C(\mu(x,q),\mu(x^{-1}y,q))\\ &=C(\mu(x,q),\mu(e_{G},q))\\ & \leq C(\mu(x,q),\mu(x,q))\\ &=\mu(x,q)\end{align*} and thus \( \mu(x,q)=\mu(y,q) .\)

Proposition 7. If \( \mu \in NAQFSC(G),\) then the set \(\frac{G}{ \mu} = \lbrace x \mu : x\in G \rbrace\) is a group with the operation \((x \mu)(y \mu) = (xy) \mu.\)

Proof.

1. If \(x,y \in G,\) then \( xy \in G. \) If \(x\mu , y\mu \in \frac{G}{\mu}\) then \( (x \mu) (y\mu)=(xy) \mu \in \frac{G}{\mu }.\)
2. Let \(x,y,z \in G\) then \( x(yz)=(xy)z. \) Now let \(x\mu, y\mu, z\mu \in \frac{G}{\mu}\) so \((x\mu) [(y\mu) (z\mu)]=[(x\mu)](yz \mu) =(xyz) \mu=(xy)z \mu=(xy \mu)(z \mu) =[(xy \mu)] (z \mu)=[(x \mu)(y\mu)](z \mu).\)
3. Let \(x \in G\) then \( xe_{G}=e_{G}x=x. \) Thus \( (x\mu)(e_{G}\mu)=(xe_{G}\mu) =(e_{G}x\mu)=x\mu.\) If \(x \in G,\) then there is an element \(x^{-1} \in G\) such that \(xx^{-1} = x^{-1}x =e_{G}.\) Let \( x\mu \in \frac{G}{\mu} \) there is an element \((x\mu)^{-1}= x^{-1}\mu \in \frac{G}{\mu}\) such that \( (x\mu)(x\mu)^{-1}=(x\mu)(x^{-1}\mu)=(xx^{-1}\mu) =(x^{-1}x\mu) =e_{G} \mu=\mu.\)

Hence \(\frac{G}{\mu}\) is a group.

Proposition 8. Let \(f : G \to H\) be a homomorphism of groups and let \( \nu \in NAQFSC(H)\) and \( \mu\) be homomorphic pre-image of \( \nu. \) Then \(\varphi : \frac{G}{\mu} \to \frac{H}{\nu}\) such that \(\varphi( x\mu ) = f(x)\nu,\) for every \(x \in G\), is an isomorphism of groups.

Proof. Let \( x, y \in G \) and \( q \in Q. \) Then \[\varphi((x\mu)(y\mu))=\varphi((xy)\mu)=f(xy)\nu=f(x)f(y)\nu=f(x)\nu f(y)\nu=\varphi(x\mu)\varphi(y\mu)\] and so \(\varphi\) is a group homomorphism. Clearly \( \varphi\) is onto and we prove that \( \varphi \) is one-one. If \( \varphi(x\mu)=\varphi(y\mu),\) then \( f(x)\nu=f(y)\nu \) and from Proposition 5 we get \[ \nu(f(x)^{-1}f(y),q) =\nu(f(y)^{-1}f(x),q)=\nu(f(e_{G}),q),\] so \[ \nu(f(x^{-1})f(y),q) =\nu(f(y^{-1})f(x),q)=\nu(f(e_{G}),q),\] then \[ \nu(f(x^{-1}y),q) =\nu(f(y^{-1}x),q)=\nu(f(e_{G}),q),\] which implies that \[\mu(x^{-1}y,q)=\mu(y^{-1}x,q)=\mu(e_{G},q),\] and \( x\mu=y\mu \) which implies that \( \varphi \) is one-one. Therefore \( \varphi \) will be an isomorphism of groups.

Proposition 9. Let \(f : G \to H\) be an anti homomorphism of groups and let \( \nu \in NQFST(H)\) and \( \mu\) be anti homomorphic pre-image of \( \nu. \) Then \(\varphi : \frac{G}{\mu} \to \frac{H}{\nu}\) such that \(\varphi( x\mu ) = f(x)\nu,\) for every \(x \in G\), is an isomorphism of groups.

Proof. Firstly, we prove that \(\varphi\) is an anti group homomorphism. Let \( x, y \in G \) and \( q \in Q. \) Then \[\varphi((x\mu)(y\mu))=\varphi((xy)\mu)=f(xy)\nu=f(y)f(x)\nu=f(y)\nu f(x)\nu=\varphi(y\mu)\varphi(x\mu),\] and so \(\varphi\) is an anti group homomorphism. Clearly \( \varphi\) is onto and we prove that \( \varphi \) is one-one. If \( \varphi(x\mu)=\varphi(y\mu),\) then \( f(x)\nu=f(y)\nu \) and from Proposition 5 we get \[ \nu(f(x)^{-1}f(y),q) =\nu(f(y)^{-1}f(x),q)=\nu(f(e_{G}),q),\] so \[ \nu(f(x^{-1})f(y),q) =\nu(f(y^{-1})f(x),q)=\nu(f(e_{G}),q),\] then \[ \nu(f(yx^{-1}),q) =\nu(f(xy^{-1}),q)=\nu(f(e_{G}),q),\] which implies that \[\mu(yx^{-1},q)=\mu(xy^{-1},q)=\mu(e_{G},q)\] and thus Proposition 5 gives us \( x\mu=y\mu \) which implies that \( \varphi \) is one-one. Therefore \( \varphi \) will be an anti isomorphism of groups.

Acknowledgments

It is my pleasant duty to thank referees for their useful suggestions which helped me to improve our manuscript.

Conflicts of Interest

The author declares no conflict of interest.