1. Introduction

A graph having no loop or multiple edge in known as simple graph. A molecular graph is a simple graph in which atoms and bonds are represented by vertices and edges respectively. The degree of a vertex is the number of edges attached with it. The maximum degree of vertex among the vertices of a graph is denoted by \(\Delta(G)\). Kulli [1] introduces the concept of reverse vertex degree \(c_{v}\), as \(c_{v}=\Delta(G)-d_{g}(v)+1\).
In discrete mathematics, graph theory in general, not only the study of different properties of objects but it also tells us about objects having same properties as investigating object [2]. In particular, graph polynomials related to graph are rich in information [3, 4, 5, 6, 7, 8].
Mathematical tools like polynomials and topological based numbers have significant importance to collect information about properties of chemical compounds [9, 10, 11]. We can find out many hidden information about compounds through theses tools.

Actually, topological indices are numeric quantities that tells us about the whole structure of graph. There are many topological indices [12, 13, 14, 15] that helps us to study physical, chemical reactivities and biological properties. Wiener [16] in 1947, firstly introduce the concept of topological index while working on boiling point. Hosoya polynomial [3] plays an important role in the area of distance-based topological indices. We can find out Wiener index, Hyper Wiener index and Tratch-stankevich-zefirove index from Hosoya polynomial. Rendić index defined by Milan Rendić [17] in 1975 is one of the oldest degree based topological index and have been extensively studied by mathematician and chemists [18, 19, 20, 21, 22]. Later Gutman et al. introduced the first and second Zagreb indices as:

$$M_{1}(G)=\sum_{uv \in E(G)}(d_{u}+d_{v}),$$ $$ M_{2}(G)=\sum_{uv \in E(G)}(d_{u}.d_{v}).$$ Zagreb indices help us in finding \(\pi\)-electronic energy [23]. Many papers [24, 25, 26, 27], surveys [23, 28, 29] and many modification of Zagreb indices are presented in literture [1, 30, 31, 32, 33, 34]. First and second Zagreb polynomials were defined in [7] as: $$M_{1}(G,x)=\sum_{uv \in E(G)}x^{(d_{u}+d_{v})},$$ $$M_{2}(G,x)=\sum_{uv \in E(G)}x^{(d_{u}.d_{v})}.$$ Motivated by these indices, Shirdel et al. [35] proposed the first and second hyper Zagreb indices as: $$H_{1}(G)=\sum_{uv \in E(G)}(d_{u}+d_{v})^{2},$$ $$H_{2}(G)=\sum_{uv \in E(G)}(d_{u}.d_{v})^{2}.$$ The first and second Reverse Zagreb indices [1] are defined as: $$CM_{1}(G)=\sum_{uv \in E(G)}(c_{u}+c_{v}),$$ $$CM_{2}(G)=\sum_{uv \in E(G)}(c_{u}.c_{v}).$$ The first and second Reverse Hyper Zagreb indices [1] are defined as: $$HCM_{1}(G)=\sum_{uv \in E(G)}(c_{u}+c_{v})^{2},$$ $$HCM_{2}(G)=\sum_{uv \in E(G)}(c_{u}.c_{v})^{2}.$$ With the help of reverse Zagreb and hyper Zagreb indices, we now able to define the reverse Zagreb and reverse hyper Zagreb polynomials as: $$CM_{1}(G,x)=\sum_{uv \in E(G)}x^{(c_{u}+c_{v})}, $$ $$CM_{2}(G,x)=\sum_{uv \in E(G)}x^{(c_{u}.c_{v})},$$ and $$HCM_{1}(G,x)=\sum_{uv \in E(G)}x^{(c_{u}+c_{v})^{2}}, $$ $$ HCM_{2}(G,x)=\sum_{uv \in E(G)}x^{(c_{u}.c_{v})^{2}}.$$ Till now there are more than 148 topological indices and non of them complete describe all properties of understudy molecular compound, so there is always room to define new topological indices. Our aim is to study Silicon Carbide \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}II[r,s]\). The multiplicative first and second Zagreb indices, multiplicative hyper Zagreb indices, and some other multiplicative degree-based topological indices of \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}II[r,s]\) are computed in [36]. Imran et al. in [37] computed the general Rendić and Zagreb types indices, geometric arithmetic index, atom-bond connectivity index, fourth atom-bond connectivity and fifth geometric arithmetic index of \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}II[r,s]\). In this report, we aim to compute reversed first and second Zagreb indices, reversed first and second Hyper Zagreb indices, reversed first and second Zagreb polynomials and reversed first and second Hyper Zagreb polynomials for \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}II(G).\) Figures of this paper are taken from [36, 37].

2. Silicon Carbide \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}II[r,s]\)

In 1891, an American scientist discover Silicon Carbide. But now a days, we can produce silicon carbide artificially by silica and carbon. Till 1929, silicon carbide was known as the hardest material on earth. Its Mohs hardness rating is 9, which makes this similar to diamond. Here, we will find out reverse zagreb, hyper reverse Zagreb and its polynomials for silicon carbide \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}II[r,s]\).

3. Main Results

Here, we will compute reverse Zagreb and reverse hyper Zagreb indices for Silicon Carbide \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}II[r,s]\), where \(r\) is the number of connected unit cells, in row (chain) and \(s\) is the number of connected rows each with \(r\) number of cells.

3.1. Silicon Carbide \(Si_{2}C_{3}I[r,s]\)

Theorem 3.1. For the Silicon Carbide \(Si_{2}C_{3}I[r,s]\), the first and second reverse Zagreb indices are:

1. \(CM_{1}(Si_{2}C_{3}I[r,s])=30rs+4r+6s-4, \)
2. \(CM_{2}(Si_{2}C_{3}I[r,s])= 15rs+7r+9s-2. \)

Proof. The vertex and edge set of Silicon Carbide is , \(|V(Si_{2}C_{3}I[r,s])|=10rs\) and \(|E(Si_{2}C_{3}I[r,s])|=15rs-2r-3s\), respectively. From figure[1-4], we can say that, there are five type of edges in \(Si_{2}C_{3}I[r,s]\). The edge set of \(Si_{2}C_{3}I[r,s]\) is portioned into the following five edge sets: $$E_{1}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); d_{u}=1, d_{v}=2\},$$ $$E_{2}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); d_{u}=1, d_{v}=3\},$$ $$E_{3}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); d_{u}=2, d_{v}=2\},$$ $$E_{4}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); d_{u}=2, d_{v}=3\},$$ $$E_{5}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); d_{u}=3, d_{v}=3\},$$ such that, \(|E_{1}(Si_{2}C_{3}I[r,s])|=1\),
\(|E_{2}(Si_{2}C_{3}I[r,s])|=1\),
\(|E_{3}(Si_{2}C_{3}I[r,s])|=r+2s\),
\(|E_{4}(Si_{2}C_{3}I[r,s])|=6r-1+8(s-1) \)
and \(|E_{5}(Si_{2}C_{3}I[r,s])|=15rs-9r-13s+7.\)
As, the maximum degree in \(Si_{2}C_{3}I[r,s]\) is 3, so, $$c_{u}=\Delta(G)-d_{G}(u)+1= 4-d_{G}(u).$$ The reverse edge set of \(Si_{2}C_{3}I[r,s]\) is given as: $$CE_{1}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); c_{u}=3, c_{v}=2\},$$ $$CE_{2}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); c_{u}=3, c_{v}=1\},$$ $$CE_{3}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); c_{u}=2, c_{v}=2\},$$ $$CE_{4}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); c_{u}=2, c_{v}=1\},$$ $$CE_{5}(Si_{2}C_{3}I[r,s])=\{uv\in E(Si_{2}C_{3}I[r,s]); c_{u}=1, c_{v}=1\}.$$ And, \(|CE_{1}(Si_{2}C_{3}I[r,s])|=1\),
\(|CE_{2}(Si_{2}C_{3}I[r,s])|=1\),
\(|CE_{3}(Si_{2}C_{3}I[r,s])|=r+2s\),
\( |CE_{4}(Si_{2}C_{3}I[r,s])|=6r-1+8(s-1)\)
and \(|CE_{5}(Si_{2}C_{3}I[r,s])|=15rs-9r-13s+7\).
(1) The first reverse Zagreb index for \(Si_{2}C_{3}I[r,s]\) is: \begin{eqnarray*} CM_{1}(Si_{2}C_{3}I[r,s])&=&\sum_{uv\in E(G)}(c_{u}+c_{v})\\ &=&(3+2)(1)+(3+1)(1)+(2+2)(r+2s)+(2+1)\\ & &(6r-1+8(s-1))+(1+1)(15rs-9r-13s+7)\\ &=&30rs+4r+6s-4. \end{eqnarray*} (2) The second reverse Zagreb index for \(Si_{2}C_{3}I[r,s]\) is: \begin{eqnarray*} CM_{2}(Si_{2}C_{3}I[r,s])&=&\sum_{uv\in E(G)}(c_{u}.c_{v})\\ &=&(3\cdot2)(1)+(3\cdot1)(1)+(2\cdot2)(r+2s)+(2\cdot1)(6r-1\\ & &+8(s-1))+(1\cdot1)(15rs-9r-13s+7)\\ &=&15rs+7r+9s-2. \end{eqnarray*}

Theorem 3.2. The first and second reverse Zagreb polynomials for \(Si_{2}C_{3}I[r,s]\) are:

1. \(CM_{1}(Si_{2}C_{3}I[r,s],x)=x^{5}+(r+2s+1)x^{4}+(6r-1+8(s-1))x^{3}+(15rs-9r-13s+7)x^{2}, \)
2. \(CM_{2}(Si_{2}C_{3}I[r,s],x)=x^{6}+(r+2s)x^{4}+x^{3}+(6r-1+8(s-1))x^{2}+(15rs-9r-13s+7)x.\)

Proof. Now, by the reverse edge partitions for \(Si_{2}C_{3}I[r,s]\), we have:
(1) The first reverse Zagreb polynomial for \(Si_{2}C_{3}I[r,s]\), is given as: \begin{eqnarray*} CM_{1}(Si_{2}C_{3}I[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}+c_{v})}\\ &=&(1)x^{(3+2)}+(1)x^{(3+1)}+(r+2s)x^{(2+2)}+(6r-1\\ & &+8(s-1))x^{(2+1)}+(15rs-9r-13s+7)x^{(1+1)}\\ &=&x^{5}+(r+2s+1)x^{4}+(6r-1+8(s-1))x^{3}\\&&+(15rs-9r-13s+7)x^{2}. \end{eqnarray*} (2) The second reverse Zagreb polynomial for \(Si_{2}C_{3}I[r,s]\), is given as: \begin{eqnarray*} CM_{2}(Si_{2}C_{3}I[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}.c_{v})}\\ &=&(1)x^{(3\cdot2)}+(1)x^{(3.1)}+(r+2s)x^{(2\cdot2)}+(6r-1\\ & &+8(s-1))x^{(2\cdot1)}+(15rs-9r-13s+7)x^{(1\cdot1)}\\ &=& x^{6}+(r+2s)x^{4}+x^{3}+(6r-1+8(s-1))x^{2}\\&&+(15rs-9r-13s+7)x. \end{eqnarray*}

Theorem 3.3. The first and second reverse hyper Zagreb indices of Silicon Carbon \(Si_{2}C_{3}I[r,s]\) are:

1. \(HCM_{1}(Si_{2}C_{3}I[r,s])=60rs+34r+50s-12,\)
2. \(HCM_{2}(Si_{2}C_{3}I[r,s])=15rs+31r+55s-4.\)

Proof. By reverse edge partition and definition of reverse hyper Zagreb indices, we have:
(1) The first reverse hyper Zagreb index for \(Si_{2}C_{3}I[r,s]\) is: \begin{eqnarray*} CM_{1}(Si _{2}C_{3}I[r,s])&=&\sum_{uv\in E(G)}(c_{u}+c_{v})^{2}\\ &=&(3+2)^{2}(1)+(3+1)^{2}(1)+(2+2)^{2}(r+2s)+(2+1)^{2}\\ & &(6r-1+8(s-1))+(1+1)^{2}(15rs-9r-13s+7)\\ &=&60rs+34r+50s-12. \end{eqnarray*} (2) The second reverse hyper Zagreb index for \(Si_{2}C_{3}I[r,s]\) is: \begin{eqnarray*} CM_{2}(Si_{2}C_{3}I[r,s])&=&\sum_{uv\in E(G)}(c_{u}.c_{v})^{2}\\ &=&(3\cdot2)^{2}(1)+(3\cdot1)^{2}(1)+(2\cdot2)^{2}(r+2s)+(2\cdot1)^{2}\\ & &(6r-1+8(s-1))+(1\cdot1)^{2}(15rs-9r-13s+7)\\ &=&15rs+31r+55s-4. \end{eqnarray*}

Theorem 3.4. The first and second reverse hyper Zagreb polynomials of \(Si_{2}C_{3}$ $I[r,s]\) are:

1. \(HCM_{1}(Si_{2}C_{3}I[r,s],x)= x^{25}+(r+2s+1)x^{16}+(6r-1+8(s-1))x^{9}+(15rs-9r-13s+7)x^{4}, \)
2. \(HCM_{1}(Si_{2}C_{3}I[r,s],x)= x^{36}+(r+2s)x^{16}+x^{9}+(6r-1+8(s-1))x^{4}+(15rs-9r-13s+7)x.\)

Proof. Now, by the reverse edge partitions for \(Si_{2}C_{3}I[r,s]\), we have
(1) The first reverse Zagreb polynomial for \(Si_{2}C_{3}I[r,s]\), is given as: \begin{eqnarray*} HCM_{1}(Si_{2}C_{3}I[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}+c_{v})^{2}}\\ &=&(1)x^{(3+2)^{2}}+(1)x^{(3+1)^{2}}+(r+2s)x^{(2+2)^{2}}+(6r-1\\ & &+8(s-1))x^{(2+1)^{2}}+(15rs-9r-13s+7)x^{(1+1)^{2}}\\ &=&x^{25}+(r+2s+1)x^{16}+(6r-1+8(s-1))x^{9}\\&&+(15rs-9r-13s+7)x^{4}. \end{eqnarray*} (2) The second reverse Zagreb polynomial for \(Si_{2}C_{3}I[r,s]\), is given as: \begin{eqnarray*} HCM_{2}(Si_{2}C_{3}I[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}.c_{v})^{2}}\\ &=&(1)x^{(3\cdot2)^{2}}+(1)x^{(3\cdot1)^{2}}+(r+2s)x^{(2\cdot2)^{2}}+(6r-1\\ & &+8(s-1))x^{(2\cdot1)^{2}}+(15rs-9r-13s+7)x^{(1\cdot1)^{2}}\\ &=& x^{36}+(r+2s)x^{16}+x^{9}+(6r-1+8(s-1))x^{4}\\&&+(15rs-9r-13s+7)x. \end{eqnarray*}

In the following Table 1, we computed first and second reverse Zagreb and first and second reverse hyper Zagreb indices for \(Si_{2}C_{3}I[r,s]\) for specific values of \(r\) and \(s\).

Table 1. first and second reverse Zagreb and first and second reverse hyper Zagreb indices for \(Si_{2}C_{3}I[r,s]\) for specific values of \(r\) and \(s\)

	\(r=1,s=1\)	\(r=1,s=2\)	\(r=2,s=1\)	\(r=2,s=1\)	\(r=2,s=3\)	\(r=3,s=3\)
\(CM_{2}\)	36	72	70	136	202	296
\(CM_{2}\)	29	53	51	90	129	181
\(HCM_{2}\)	132	242	226	396	566	780
\(HCM_{2}\)	97	167	143	228	313	389

3.2. Silicon Carbide \(Si_{2}C_{3}II[r,s]\)

Theorem 3.5. For the Silicon Carbide \(Si_{2}C_{3}II[r,s]\), the first and second reverse Zagreb indices are:

1. \(CM_{1}(Si_{2}C_{3}II[r,s])=30rs+6r+6s-6 ,\)
2. \(CM_{2}(Si_{2}C_{3}II[r,s])= 15rs+11r+11s-2. \)

Proof. The vertex and edge set of Silicon Carbide is, \(|V(Si_{2}C_{3}II[r,s])|=10rs\) and \(|E(Si_{2}C_{3}II[r,s])|=15rs-2r-3s\), respectively. From the Figures 5-8, we can observe that, there are five type of edges in \(Si_{2}C_{3}II[r,s]\). The edge set of \(Si_{2}C_{3}II[r,s]\) is portioned into following five edge sets: $$E_{1}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); d_{u}=1, d_{v}=2\},$$ $$E_{2}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); d_{u}=1, d_{v}=3\},$$ $$E_{3}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); d_{u}=2, d_{v}=2\},$$ $$E_{4}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); d_{u}=2, d_{v}=3\},$$ $$E_{5}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); d_{u}=3, d_{v}=3\},$$ such that, \(|E_{1}(Si_{2}C_{3}II[r,s])|=2\),
\( |E_{2}(Si_{2}C_{3}II[r,s])|=1\),
\(|E_{3}(Si_{2}C_{3}II[r,s])|=2r+2s\),
\(|E_{4}(Si_{2}C_{3}II[r,s])|=8r+8s-14\)
and \(|E_{5}(Si_{2}C_{3}II[r,s])|=15rs-13r-13s+11.\)
The maximum vertex degree \(Si_{2}C_{3}II[r,s]\) is 3, so, $$c_{u}=\Delta(G)-d_{G}(u)+1= 4-d_{G}(u).$$ The reverse edge set of \(Si_{2}C_{3}II[r,s]\) is given as: $$CE_{1}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); c_{u}=3, c_{v}=2\},$$ $$CE_{2}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); c_{u}=3, c_{v}=1\},$$ $$CE_{3}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); c_{u}=2, c_{v}=2\},$$ $$CE_{4}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); c_{u}=2, c_{v}=1\},$$ $$CE_{5}(Si_{2}C_{3}II[r,s])=\{uv\in E(Si_{2}C_{3}II[r,s]); c_{u}=1, c_{v}=1\},$$ and we have, \(|E_{1}(Si_{2}C_{3}II[r,s])|=2\),
\(|E_{2}(Si_{2}C_{3}II[r,s])|=1\),
\( |E_{3}(Si_{2}C_{3}II[r,s])|=2r+2s\),
\(|E_{4}(Si_{2}C_{3}II[r,s])|=8r+8s-14\)
and \(|E_{5}(Si_{2}C_{3}II[r,s])|=15rs-13r-13s+11\).
(1) The first reverse Zagreb index for \(Si_{2}C_{3}II[r,s]\) is: \begin{eqnarray*} CM_{1}(Si_{2}C_{3}II[r,s])&=&\sum_{uv\in E(G)}(c_{u}+c_{v})\\ &=&(3+2)(2)+(3+1)(1)+(2+2)(2r+2s)+(2+1)\\ & &(8r+8s-14)+(1+1)(15rs-13r-13s+11)\\ &=&30rs+6r+6s-6. \end{eqnarray*} (2) The second reverse Zagreb index for \(Si_{2}C_{3}II[r,s]\) is: \begin{eqnarray*} CM_{2}(Si_{2}C_{3}II[r,s])&=&\sum_{uv\in E(G)}(c_{u}.c_{v})\\ &=&(3\cdot2)(2)+(3\cdot1)(1)+(2\cdot2)(2r+2s)+(2\cdot1)\\ & &(8r+8s-14)+(1\cdot1)(15rs-13r-13s+11)\\ &=&15rs+11r+11s-2. \end{eqnarray*}

Theorem 3.6. The first and second reverse Zagreb polynomials for \(Si_{2}C_{3}II[r,s]\) are:

1. \(CM_{1}(Si_{2}C_{3}II[r,s],x)= x^{5}+(2r+2s+1)x^{4}+(8r+8s-14)x^{3}+(15rs-9r-13s+7)x^{2},\)
2. \(CM_{2}(Si_{2}C_{3}II[r,s],x)=2x^{6}+(2r+2s)x^{4}+x^{3}+(8r+8s=14)x^{2}+(15rs-13r-13s+11)x.\)

Proof. Now, by the reverse edge partitions for \(Si_{2}C_{3}II[r,s]\), we have:
(1) The first reverse Zagreb polynomial for \(Si_{2}C_{3}II[r,s]\), is given as: \begin{eqnarray*} CM_{1}(Si_{2}C_{3}II[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}+c_{v})}\\ &=&(2)x^{(3+2)}+(1)x^{(3+1)}+(2r+2s)x^{(2+2)}+(8r\\ & &+8s-14)x^{(2+1)}+(15rs-13r-13s+11)x^{(1+1)}\\ &=&x^{5}+(2r+2s+1)x^{4}+(8r+8s-14)x^{3}\\&&+(15rs-9r-13s+7)x^{2}. \end{eqnarray*} (2) The second reverse Zagreb polynomial for \(Si_{2}C_{3}II[r,s]\), is given as: \begin{eqnarray*} CM_{2}(Si_{2}C_{3}II[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}.c_{v})}\\ &=&(2)x^{(3\cdot2)}+(1)x^{(3\cdot1)}+(2r+2s)x^{(2\cdot2)}+(8r\\ & &+8s-14)x^{(2\cdot1)}+(15rs-13r-13s+11)x^{(1.\cdot1)}\\ &=& 2x^{6}+(2r+2s)x^{4}+x^{3}+(8r+8s-14)x^{2}\\&&+(15rs-13r-13s+11)x. \end{eqnarray*}

Theorem 3.7. The first and second reverse Hyper Zagreb indices of Silicon Carbon \(Si_{2}C_{3}II[r,s]\) are:

1. \(HCM_{1}(Si_{2}C_{3}II[r,s])=60rs+51r+51s-16,\)
2. \(HCM_{2}(Si_{2}C_{3}II[r,s])=15rs+51r+51s+108.\)

Proof. By reverse edge partition and definition of reverse hyper Zagreb indices, we have:
(1) The first reverse Hyper Zagreb index for \(Si_{2}C_{3}II[r,s]\) is: \begin{eqnarray*} CM_{1}(Si_{2}C_{3}II[r,s])&=&\sum_{uv\in E(G)}(c_{u}+c_{v})^{2}\\ &=&(3+2)^{2}(2)+(3+1)^{2}(1)+(2+2)^{2}(2r+2s)+(2+1)^{2}\\ & &(8r+8s-14)+(1+1)^{2}(15rs-13r-13s+11)\\ &=&60rs+52r+52s-16. \end{eqnarray*} (2) The second reverse Hyper Zagreb index for \(Si_{2}C_{3}I[r,s]\) is: \begin{eqnarray*} CM_{2}(Si_{2}C_{3}II[r,s])&=&\sum_{uv\in E(G)}(c_{u}.c_{v})^{2}\\ &=&(3+2)^{2}(2)+(3+1)^{2}(1)+(2+2)^{2}(2r+2s)\\ & &+(2+1)^{2}(8r+8s-14)+(1+1)^{2}(15rs\\&&-13r-13s+11)\\ &=&15rs+51r+51s+108. \end{eqnarray*}

Theorem 3.8. The first and second reverse hyper Zagreb polynomials of \(Si_{2}C_{3}$ $II[r,s]\) is:

1. \(HCM_{1}(Si_{2}C_{3}II[r,s],x)= 2x^{25}+(2r+2s+1)x^{16}+(8r+8s-14)x^{9}+(15rs-13r-13s+11)x^{4}, \)
2. \(HCM_{1}(Si_{2}C_{3}II[r,s],x)= 2x^{36}+(2r+2s)x^{16}+x^{9}+(8r+8s-14)x^{4}+(15rs-13r-13s+11)x.\)

Proof. Now, by the reverse edge partitions for \(Si_{2}C_{3}II[r,s]\), we have:
(1) The first reverse Zagreb polynomial for \(Si_{2}C_{3}II[r,s]\), is given as: \begin{eqnarray*} CM_{1}(Si_{2}C_{3}II[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}+c_{v})^{2}}\\ &=&(2)x^{(3+2)^{2}}+(1)x^{(3+1)^{2}}+(2r+2s)x^{(2+2)^{2}}+(8r\\ & &+8s-14)x^{(2+1)^{2}}+(15rs-13r-13s+11)x^{(1+1)^{2}}\\ &=&2x^{25}+(2r+2s+1)x^{16}+(8r+8s-14)x^{9}\\&&+(15rs-13r-13s+11)x^{4}. \end{eqnarray*} (2) The second reverse Zagreb polynomial for \(Si_{2}C_{3}I[r,s]\), is given as: \begin{eqnarray*} CM_{2}(Si_{2}C_{3}II[r,s],x)&=&\sum_{uv\in E(G)}x^{(c_{u}.c_{v})^{2}}\\ &=&(2)x^{(3\cdot2)^{2}}+(1)x^{(3\cdot1)^{2}}+(2r+2s)x^{(2\cdot2)^{2}}+(8r+8s\\ & &-14)x^{(2\cdot1)^{2}}+(15rs-13r-13s+11)x^{(1\cdot1)^{2}}\\ &=&2x^{36}+(2r+2s)x^{16}+x^{9}+(8r+8s-14)x^{4}\\&&+(15rs-13r-13s+11)x. \end{eqnarray*}

In the following Table 2, we computed first and second reverse Zagreb and first and second reverse Hyper Zagreb indices for \(Si_{2}C_{3}II[r,s]\) for spefic values of \(r\) and \(s\).

Table 2. first and second reverse Zagreb and first and second reverse Hyper Zagreb indices for \(Si_{2}C_{3}II[r,s]\) for spefic values of \(r\) and \(s\)

	\(r=1,s=1\)	\(r=1,s=2\)	\(r=2,s=1\)	\(r=2,s=1\)	\(r=2,s=3\)	\(r=3,s=3\)
\(CM_{2}\)	36	72	72	138	204	300
\(CM_{2}\)	35	61	61	162	143	199
\(HCM_{2}\)	146	257	257	428	599	830
\(HCM_{2}\)	225	291	291	372	453	549

Conclusion

\noindent The first and second Zagreb indices are used to compute total \(\pi\)-energy of conjugated molecules. These indices are also useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. In this paper we have obtained reverse Zagreb indices, hyper reverse Zagreb indices and their polynomials for Silicon Carbide \(Si_{2}C_{3}I[r,s]\) and \(Si_{2}C_{3}I[r,s]\).

Competing Interests

The authors declare that they have no competing interests.

Nicotiana tabaccum, is a stout herbaceous plant in the solanaceae family that originated in the tropical Americas and now cultivated worldwide [1]. It is the commercial source of tobacco. It contain nicotine, a powerful neurotoxin that is particularly harmful to insects [2]. Nicotine, a carcinogen, is the compound responsible for the addictive nature of tobacco use [3]. Nicotine is also used as insecticides. Structure of nicotine is shown in Figure 1.

It was observed by literature that nicotine could be extracted by different techniques. Nicotine and zinc are closely related to a variety of brain pathologies, including schizophrenia, anxiety, depression, Parkinson's and Alzheimer's diseases [4]. Yuegang Zuo developed ultrasonic extraction (UE) of nicotine with heptane followed by direct capillary gas chromatography (GC) separation [5]. Lubomir Svorc developed a sensitive, selective and reliable electrochemical method for the determination of nicotine using differential pulse voltammeter on a bare boron-doped diamond electrode [6]. We have used the technique which is not used in previous studies. Our method is more economical and convenient. Nicotine was isolated from the seeds of N. tabacum and its Zinc (II) complex was synthesized [7] and now in the present study, nicotine is isolated from different brands of cigarettes and its Zinc complexes are synthesized and bacterial sensitivity against different micro-organisms were performed. The outcomes of the study reveal that Zinc-Nicotine complexes can be used as therapeutic agents and anti-sickness agents in sickle cell disease.

2. Materials and Methods

Aluminum sheets pre-coated with silica gel (\(60 F_{254}\)), chloroform, ether, distilled water, saturated picric acid solution, Methanol, 5% NaOH solution, \(ZnCl_{2}\), and dimethylsulphoxide, R.B flask, separating funnel, Whatman filter paper no.1, are used throughout the study. Label information of five different brands of cigarette is given in Table 1.

Table 1. Label information of five different brands of cigarette

Sr.No.	BrandName	Abbreviation	Company	Price / Cig	Total Price/20cig
1	Prime	Nic-A	Flacon CigaretteIndustry	2	42
2	Cafe	Nic-B	Universal TobaccoCompany	2.5	50
3	Classic	Nic-C	National TobaccoIndustries	3	58
4	Dunhill	Nic-D	Dunhill Tobacco ofLondon Limited	7.5	150

3. Experimental Work

In-vitro evaluation of nicotine is carried out from different brands of cigarettes in local market of Lahore, Pakistan. The simple extraction method is subject to isolate the nicotine alkaloid from its source.

3.1. Isolation of Nicotine from Cigarette

10g of cigarette leaves are taken in a beaker, added 100mL of 5% NaOH solution and stirred it for half hour on hot plate. Filtered it by using glass wool and transferred the cigarettes again to beaker. Added 30mL distilled water and stirred and filtered again. Collected the filtrates together and transferred to separating funnel. Extracted the nicotine by 30mL ether and repeated the extraction 3 times. Gathered the four filtrates in a conical flask and evaporated ether on water bath. Added 4mL methanol to dissolve the resulted oil. Pure nicotine is derived subsequently and then dissolved in another solvent such as ether and small amount of n-hexane was added for recrystallization.

3.2.Characterization of Pure Nicotine

Digital melting point apparatus is used to determine melting points of different extracted pure samples (after SPE Purification). The average melting point of the samples was \(-79C^0.\) Thin layer chromatography is subjected to check the specificity and purity of nicotine in all brands by using solvent system n-butanol: acetic acid: water (3:1:1 v/v) and compared it by Co-TLC with the standard. It was found that the nicotine in all the brands showed single spot (\(R_f\) values 0.47) after visualization on TLC plate which confirmed the purity and specificity of the nicotine and it is also noted that no interference of any component on TLC observed (Table 2). FT-IR spectroscopy was used for IR-spectrum of extracted purified nicotine i.e. shown in Table 3. By using an UV-absorption spectrophotometer, absorption spectrum of extracted purified crystalline nicotine is taken at different absorbance against different wavelength. The \(\lambda_{\max}\) was found to be 760nm.

Table 2. Characteristics of purified Nicotine isolated from Cigarettes

Sample	Melting Point(\(C^{0}\))	\(\lambda_{max(mn)}\)	Rf
Nic-A	-80	759	0.46
Nic-B	-78	758	0.48
Nic-C	-79	760	0.47
Nic-D	-79	760	0.47

Table 3. Characteristics IR-absorption bands of samples

Bond	v(\(cm^{-1}\))Nic-A	v(\(cm^{-1}\))Nic-B	v (\(cm^{-1}\))Nic-C	v(\(cm^{-1}\))Nic-D
CH (Stretching)	2964	2963	2968	2970
C=N	1673	1674	1676	1677
C=C	1689	1689	1690	1691
CH (Bending)	902	901	903	904

3.3. Complex Formation

Nicotine isolated from leaves of Nicotiana tabacum is used for complexation with \(Zn\) and is studied for their antibacterial activities against different strains of gram positive and gram-negative bacteria compared it with Zinc salt (\(ZnCl_{2}\)) used for complexation and nicotine alone.

3.3.1. Complex Formation of Nicotine

0.1g of sample is dissolved in 15mL methanol in round bottom flask and stirred for 5 minutes. Then 0.5g of metal salt \(ZnCl_{2}\) is dissolved in it. Then refluxed it for one hour, and separated it by adding 10mL of chloroform in separating funnel. Separated the lower layer of chloroform and placed it for evaporation. Fine and pure crystals of metal complex are obtained. Structure of zinc-nicotine complex is shown in Figure 2.

3.3.2. Identification of Complex Formation

TLC is checked to confirm the synthesis of complex compounds Nic-A, Nic-B, Nic-C and Nic-D. The compounds Nic-A and Nic-B did not show a single spot on precoated aluminum silica gel plate which confirmed the impurities in the sample A and B. It is concluded that no complex formation is done by these compounds. The compounds Nic-C and Nic-D showed single spot on TLC cards which confirmed the complex formation by these compounds.

4. Antibacterial Activity of Nicotine

4.1. Preparation of stock solutions

Stock solutions of all the test samples in the concentration of 1mg/mL are prepared in dimethylsulphoxide and then diluted to 100 and 200µg/mL with the same solvent.

4.2. Inhibition assay

The anti-microbial sensitivity tests of the title compounds are tested against ten different species of gram positive and gram-negative bacteria. The compounds are used in two concentrations that is, 100 \(\mu\)g/100 \(\mu\)L (first dose level) and 200 \(\mu\)g/100 \(\mu\)L, the second dose level. The anti-bacterial activities of nicotine and its zinc complex are determined by 'Agar Well Diffusion Method' proposed by [8]. According to this method, the weighed components of the dehydrated medium are dissolved in distilled water and made the volume to one liter. Solution was heated till boiling for complete dissolution of the components. The medium is autoclaved at the pressure of 15 Ibs/in2 for 15 min while keeping the temperature of 121°C. The autoclaved medium is then poured in the sterile Petri plates and was allowed to solidify in the clean environment. Then these plates are incubated at 37°C for 24 h to check their sterility.

4.3. Measurement of Antibacterial Activity

One loop, full of 24 h old bacterial culture containing approximately 104 to 106 CFU, was spread on the surface of Mueller-Hinton agar plates. The composition of Mueller-Hinton agar medium is given in Table 1. Wells were dug in the medium with the help of sterile metallic borer. The marked area is filled with diluted solutions of the test samples, metal salts and solvent dimethylsulphoxide. These plates are incubated at 37°C for 24 h. At the end of the incubation period, the inhibition zones are measured to the nearest millimeters. Antibacterial activity is indicated by a clear zone encircling the marked area. Beyond the marked area, there is a homogenous confluent lawn of bacterial growth.

5. Results and Discussion

In our present study, four different brands of cigarettes i.e., Nic-A, Nic-B, Nic-C and Nic-D of four different companies W, X, Y and Z are used. The basic compound (Nicotine) in all the cigarettes was same with slight variation in complexation. Our first attempt was to extract and purify the nicotine alkaloid by using simple solvent extraction method and compare them to each other and with standard for qualitative analysis. Using the experimental observations, it is found that Nic-A is extracted by weight as 1.75gm in amorphous powder; Nic-B is extracted by weight as 1.46gm, Nic-C 1.07gm and Nic-D 0.87gm as fine crystals respectively.

Further the compounds are used for metal complexes, four of which, two compounds Nic-A and Nic-B could not be subjected for complex formation because of impurities were present in them, (checked by TLC with the standard). The compounds Nic-C and Nic-D are used for Zn-nicotine complex due to its purity and fine crystals. From the results and plots it is also observed and concluded that the products of company Y and Z have been better qualified than the products of company W and X. In nicotine structure due to the presence of nitrogen atom it has unsymmetrical divergent ligand property.

Here only the antimicrobial properties are discussed with reference to the standard. The antimicrobial activity of Zn-II-nicotine complex is tested against Gram negative and Gram-positive bacteria. The concentrations of the compounds are used that is 100\(\mu\)g /100\(\mu\)L and 200$\mu$g /100\(\mu\)L for first dose level and second dose level. Results of Zn (II)-nicotine complex for both the concentrations are given in table no. V.

6. Qualitative Analysis of Nicotine

The weight of Nicotine in four different brands of cigarettes is given in Table 4 and comparison between amounts of nicotine in different brands is shown in Figure 3.

Table 4. The weight of Nicotine in four different brands of cigarettes

Sample	Amount of sample(g)	Amount ofNicotineafter solventextraction(mg)	%ageof Nicotinebefore SPE	%ageof Nicotineafter SPE
Nic-A	10	1.75	0.0175	0.0125
Nic-B	10	1.46	0.0146	0.0100
Nic-C	10	1.07	0.0107	0.0098
Nic-D	10	0.87	0.0087	0.0077

Table 5. Antimicrobial activity of Zn-II-nicotine complex

Sr.No.	Compound used	Conc. µg / 100 µL	Gram positive organisms	Gram negative organisms
1	Gentamycin	100200	3140	3580
2	Tetracycline	100200	1225	0.0125
3	Tobramycin	100200	3070	3050
4	Nicotine	100200	14	0.0098
5	Zn-II-nicotinecomplex	100200	2027	1820

The results reveal that nicotine is inactive for 1st dose level and is more effective antibiotic against 2nd dose level for Gram positive and Gram-negative bacteria i.e. S. faecalis and E. coli respectively. Zn-nicotine complex did not inhibit the both Gram negative and Gram-positive bacteria however, in the 2nd dose level; it inhibited the growth of both the bacterial species. The results are also compared with three antibiotics that is Gentamycin, Tetracycline, Tobramycin. On the basis of above evidences, we comment that this Zn-nicotine complex is broad applicant antimicrobial agent and is active against Gram positive and Gram-negative bacterial species. Further, Zinc-Nicotine complexes can be used as therapeutic agents and anti-sickness agents in sickle cell disease.

Competing Interests

The authors declare that they have no competing interests.

1. Introduction

Nanotechnology is an escalating field in recent science [1] that controls matter which is extremely small (< 100nm). Idea of Nanotechnology was proposed by Richard Feynman [2] and Professor Norio Taniguchi coined this term in 1974. Nanotechnology is the segregation, organization, and strengthening of materials. Nanoparticles are ultrafine particles [3] with exceptional physico-chemical characteristics which are the result of quantum confinement effect [4]. In particles of the size less than 70nm Vander waals force becomes significant. Gecko can climb the walls due to nanosized hairs on its limbs. Gold NPs change their ability to reflect light and Al NPs are extremely reactive when their size is less than 20nm. Nanoparticles are fascinating because of their change of properties when they are very small [5]. Concept of Nanoparticles is not new, they have been in use since 4th century A.D, for example Lycurgus cup [6]. Nanoparticles have various applications in catalysis, electronic devices, dyes and pigments [6, 7].

Physical, chemical, and biological methods are commonly used for the synthesis of nanoparticles. Physical methods are free from contaminants but are uneconomical because of formation of plentiful waste. Physical methods used for the synthesis of nanoparticles are ball-milling, ablation, and pyrolysis. Various chemical methods used for the synthesis of nanoparticles are sol-gel method, microemulsion, hydrothermal, and chemical vapor deposition. Biological methods are clean and environment friendly methods. Various biological entities like fungi, bacteria, and plants are used in this method. This method involves enzymes, proteins, and NADH reductase coenzyme [8].

Biological methods using microorganisms are clean, nonhazardous, and eco friendly. These are fast and are carried out at ordinary conditions. Microorganisms are adapted to harsh conditions by using survival strategies such as efflux system, extracellular precipitation and chemical detoxification. Reduction, biosorption, and bioaccumulation are important mechanisms for biosynthesis [9, 10].

Chromium (iii) oxide nanoparticles are one of the unique transition metal compounds [11] that have won much attention of researchers because of their extensive use in science and technology [12]. The \(Cr_2O_3\) nanoparticles are manufactured by an aqueous precipitation method using chromic sulphate as a template and ammonia as a precipitating agent. Gibot and Vidal synthesized spherical \(Cr_2O_3\) nanoparticles by thermal decomposition of \(Cr (NO_3)_3.9H_2O\) [13]. Pei et al. synthesized \(Cr_2O_3\) nanoparticles by using \(CrO_3\) and \(C_2H_5OH.\) Jaswal et al. synthesized \(Cr_2O_3\) NPs by precipitation method using chromic sulphate and ammonia. This method is of low cost and environment friendly [14].

\(Cr_2O_3\) NPs are fabricated by reducing potassium dichromate solution with Tridaxprocumbens leaf extract [15] and Allium sativum [16]. Pure and uniform sized \(Cr_2O_3\) NPs with cubic morphology were synthesized by a green chemistry method using Callistemon viminalis flowers extract [17]. Chromium oxide NPs were synthesized by mixing potassium dichromate solution with pumpkin leaves extract. The solution was dried at 70°C for 6 hours and then calcined at 650°C [18]. The use of bioorganisms is ecofriendly approach with natural reducing and capping agents [19]. Chromium oxide NPs find wide range of applications in colorants, catalysts, coatings [20], green pigment, solar energy collectors, and liquid crystal displays [21].

2. Experimental

In present study we synthesize \(Cr_2O_3\) NPs by using biological method. This study is carried out at Nanoscience and Technology Department, National Centre for Physics, Quaid-e-Azam University, Islamabad and Department of Chemistry, University of Wah, Wah-Cantt. These nanoparticles are characterized by using X-ray diffraction, UV-Visible spectroscopy, Scanning Electron Microscopy and Electron Dispersive X-ray Spectroscopy. During this work all chemicals are purchased from local market of Sigma-Aldrich. These were AR-Grade and there was no need of further purification. We used deionized water throughout the experiment.

2.1. Biological Synthesis

In this method first of all salt solution is prepared by dissolving 3-4g of salt (\(Cr_2 (SO_4)_3\)) in deionized water and mixed with 2g crushed powder of Aspargillusniger and stirred for 30 minutes. Then the resultant material ia placed in dark for 3 days. After 3 days we filtered the solution. The filtrate is characterized by UV-Visible for finding size and concentration of nanoparticles. The filtrate is then dried and calcined in furnace at 550°C for 3 hours. Now the material is grinded and analyzed by XRD, SEM and EDX.

We analyzed nanoparticles by using XRD model D8 ADVANCE BRUKER X-Source Copper/(anode). UV-Vis is performed on UV-Vis Spectrometer Perkin Elmer, Lambda 25. Both instruments are placed at Nanoscience and Technology Department, Quaid-e-Azam University, Islamabad. The Scherrer formula is used for finding size. The formed NPs are characterized by XRD and their results arenoted in nanometer. The synthesized NPs are also characterized by SEM performed on SEM, TESCAN, VEGA3 placed at Advanced Energy and Material lab NUST. The SEM study is carried out to find size and morphology of nanoparticles. The EDX is done on EDX Oxford placed at Fracture Mechanics and Fatigue Lab, Mechanical Engineering Department, UET Taxila. The EDX is used to find elemental composition and purity of samples.

3. Results and discussions

We characterized NPs by XRD. The XRD is used to find size of particles and crystallinity. The Scherrer formula is used to find crystallite size of NPs. The XRD analysis of biologically synthesized Cr2O3 nanoparticles is described Figure 1.

The peaks for chromium oxide NPs are formed at 2\(\theta\) values of 24°,33°,36°,41°,50°,55°,65° and 65°. These results were matched with JCPDF # 51-0959. The peaks are sharp and clearly distinguishable. (Table 1)

Table 1. XRD Data of \(Cr_2O_3\) NPs

PEAKS	2θ POSITION	hkl VALUES	d-SPACING
1	24.720	012	3.5986
2	33.912	104	2.6412
3	36.530	110	2.4578
4	41.865	113	2.1560
5	50.696	024	1.7992
6	55.381	116	1.6576
7	65.760	300	1.4189

SEM is used to study surface morphology of NPs by scanning the surface with high energy electrons. SEM study shows that \(Cr_2O_3\) NPs are beautiful white hexagonal crystals (Figure 2). All the crystals are uniform sized with average crystalline size of 66nm.

Chromium oxide NPs showed exciton absorbance around 421nm which was matched with the literature. The second peak at around 587nm is may be due to the presence of impurities or transition state (Figure 3). The EDX analysis confirmed the presence of of \(Cr_2O_3\) NPs.It showed 73% Cr and 27.0% oxygen content. The elemental analysis showed that chromium oxide NPs are highly pure without any trace of impurities (Figure 4).

4. Conclusion

The present study shows that Chromium oxide NPs were successfully synthesized by biological method using fungal extract. The XRD study shows that the size of biologically synthesized \(Cr_2O_3\) NPs is 36nm. The SEM study shows that \(Cr_2O_3\) NPs are hexagonal. The EDX shows that synthesized nanoparticles are pure and there is only trace of impurities present in the samples. Chromium oxide NPs have applications in the stropping of knives, glasses, inks, paints and precursor to the magnetic pigment.

Acknowledgement

We acknowledge the provision of services of NS & TD, NCP, QAU Islamabad.

Competing Interests

The authors do not have any competing interests in the manuscript.

1. Introduction

The boundedness of Littlewood-Paley operators on function spaces are one of the very important tools, not only in harmonic analysis, but also in potential theory and in partial differential equations (see [1, 2, 3, 4, 5, 6], for details). In 2004, Ding, Lin and Shao [7] investigated the \({L}^{2}\)-boundedness for a class of Marcinkiewicz integral operators with variable kernels \(\mu_{\Omega}\) and \(\mu_{\Omega,s}\) related to the Littlewood-Paley function \(\mu^{*}_{\Omega,\lambda}\) and the area integral \(g^{*}_{\lambda}\). In 2006, the authors [8] proved the \(L^{p}\)-boundedness of the Littlewood-Paley operators with variable kernels. In 2009, Xue and Ding [9] established the weighted estimate for Littlewood-Paley operators and their commutators.

In 1960, Hörmander [10] introduced the parameterized Littlewood-Paley operators for the first time. Now, let us recall the definitions of the parameterized Lusin area integral and Littlewood-Paley \(g^{*}_{\lambda}\) function. Let \(S^{n-1} (n\geq 2)\) be the unit sphere in \(\mathbb{R}^{n}\) with normalized Lebesgue measure \(\mathrm{d}\sigma(x')\). Take \(\Omega(x,z)\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq{1})\) to be a homogeneous function of degree zero and where \(\Omega\) satisfies the following conditions:

1. For any \(x,z \in{\mathbb{R}^{n}}\) and any \(\lambda > 0\), we have \(\Omega{(x,\lambda{z})}=\Omega(x,z)\);
2. \(\|\Omega\|_{L^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}}:=\sup\limits_{r\geq0, y\in\mathbb{R}^{n}}\left(\int_{s^{n-1}}|{\Omega(rz'+y ,z')}|^{r}\mathrm{d}{\sigma}(z')\right)^{\frac{1}{r}}{< \infty}.\)

The parameterized Littlewood-Paley operators \(\mu^{\rho}_{\Omega,s}\) and \(\mu^{*,\rho}_{\Omega,\lambda}\) with variable kernels, which are related to the Lusin area integral and the Littlewood-Paley \(g_{\lambda}^{*}\) function are defined by $$ \mu^{\rho}_{\Omega,s}(f)(x)=\left(\int \int_{\Gamma(x)}\left|\frac{1}{t^{\rho}}\int_{|y-z|\leq t}\frac{\Omega(y,y-z)}{|y-z|^{n-\rho}}f(z)\mathrm{d}z\right|^{2}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\right) ^{\frac{1}{2}} $$ and \begin{eqnarray*} &&\mu^{*,\rho}_{\Omega,\lambda}(f)(x)=\\&&\left(\int \int_{\mathbb{R}^{n+1}_{+}}\left(\frac{t}{t + |x-y|}\right)^{\lambda n}\left|\frac{1}{t^{\rho}}\int_{|y-z|\leq t}\frac{\Omega(y,y-z)}{|y-z|^{n-\rho}}f(z)\mathrm{d}z\right|^{2}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\right)^{\frac{1}{2}}, \end{eqnarray*} where \(\Gamma(x)= \{ (y,t)\in {\mathbb{R}^{n+1}_{+}}: |x-y|< t\) and \(\lambda >1 \}\).

In 2013, Wei and Tao [11] investigated the boundedness of parameterized Littlewood Paley operators on weighted weak Hardy spaces. Lin and Xuan [12] established the boundedness for commutators of parameterized Littlewood-Paley operators and area integrals on weighted Lebesgue spaces \(L^{p}(w)\).

The theory of the variable exponent function spaces has been rapidly developed after the work [13], where Kováčik and Rákosník have clarified fundamental properties of Lebesgue spaces with variable exponent. After that, many researchers have been interested in the theory of the variable exponent spaces (see [14, 15, 16, 17, 18, 19, 20]).

The generalization of the Muckenhoupt weights with variable exponent \(A_{p(\cdot)}\) has been considered in [21, 22, 23, 24]. The equivalence between the Muckenhoupt condition and the boundedness of the Hardy-Littlewood maximal operator on weighted Lebesgue spaces with variable exponent were discussed in [21, 22]. After that, Cruz-Uribe and Wang [25] proved the boundedness of some classical operators on weighted Lebesgue spaces with variable exponent \(L^{p(\cdot)}(w)\).

Recently, Izuki and Noi [26] introduced the weighted Herz spaces with variable exponent, and also studied the boundedness of fractional integrals on those spaces.

In this paper, we establish the boundedness of parameterized Littlewood-Paley operators with variable kernels on weighted Herz spaces with variable exponent \(\dot{K}_{p(\cdot)}^{\alpha,q}(w)\). Let \(E\) be a Lebesgue measurable set in \(\mathbb{R}^{n}\) with measure \(|E|>0\), \(\chi_{E}\) means its characteristic function. We shall recall some definitions.

Definition 1.1. Let \(p(\cdot): E \rightarrow {[1,\infty)}\) be a measurable function. The variable exponent Lebesgue space is defined as $$ L^{p(\cdot)}(E)= \{{ f~\mbox{is measurable}: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)} \mathrm{d}x <\infty}~ \mbox{for some constant } \eta > 0\}. $$ The space \(L _{\mathrm{loc}}^{p(\cdot)} {(E)}\) is defined as $$ L_{\mathrm{loc}}^{p(\cdot)} {(E)}= \{f \mbox { is measurable}: f\in {L^{p(\cdot)} {(K)}}~\mbox{for all compact} ~K\subset E\}. $$ The Lebesgue spaces \(L^{p(\cdot)} {(E)}\) is a Banach spaces with the norm defined as $$ \|f\|_{L^{p(\cdot)}(E)}= \inf\left\{\eta> 0: \int_{E}\left(\frac{|f(x)|}{\eta}\right)^{p(x)}\mathrm{d}x \leq 1\right\}. $$

We denote \(p_{-}= \mbox{ess inf} \{p(x): x \in E\},\) \(p_{+}= \mbox{ ess sup} \{p(x): x \in E\}\), then \(\mathcal{P}(E)\) consists of all \(p(\cdot)\) satisfying \(p_{-} > 1\) and \(p_{+} < \infty\).

Definition 1.2.[27] Let \(p(\cdot):\mathbb{R}^{n}\rightarrow [1,\infty)\). A measurable function \(p(\cdot)\) is said to be globally log-Höder continuous if it satisfies

1. \(|p(x)-p(y)|\leq \frac{1}{\mathrm{-log}(|x-y|)}, ~~~~~~x,y\in \mathbb{R}^{n}, |x-y|\leq 1/2;\)
2. \(|p(x)-p_{\infty}|\leq \frac{1}{\mathrm{log}(e+|x|)}, ~~~~~~~~~x\in \mathbb{R}^{n},\)

for some \(p_{\infty}\geq 1\). The set of \(p(\cdot)\) satisfying conditions (1) and (2) is denoted by \(LH(\mathbb{R}^{n})\).

We know that, if \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\), the Hardy-Littlewood maximal operator \(M\) is bounded on \(L^{p(\cdot)}(\mathbb{R}^{n})\) (see [28]).

Definition 1.3.[29] Suppose that \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and \(w\) is a weight function. The weighted Lebesgue spaces with variable exponent \(L^{p(\cdot)}(w)\) is the set of all complex-valued measurable function \(f\) such that \(fw^{1/p(\cdot)}\in L^{p(\cdot)}(\mathbb{R}^{n})\). The space \(L^{p(\cdot)}(w)\) is a Banach space equipped with the norm $$ \|f\|_{L^{p(\cdot)}(w)}= \|fw^{1/p(\cdot)}\|_{L^{p(\cdot)}}. $$ \(p'(\cdot)\) is the conjugate of \(p(\cdot)\) such that \(\frac{1}{ p'(\cdot) }+\frac{1}{ p(\cdot) }=1\). Next, we introduce the classical Muckenhoupt \(A_{p}\) weight.

Definition 1.4.[30] Let \(1< p< \infty\), then \(w\in A_{p}\) for every cube \(Q\), $$ \left( \frac{1}{|Q|}\int_{Q}w(x)\mathrm{d}x\right)\left( \frac{1}{|Q|}\int_{Q}w(x)^{1-p'}\mathrm{d}x\right)^{p-1}\leq C< \infty. $$ We say that \(w\in A_{1}\) if it satisfies \(Mw(x)\leq w(x)\) for all \(x\in \mathbb{R}^{n}\). The set \(A_{1}\) consists of all Muckenhoupt \(A_{1}\) weights.

Definition 1.5.[21, 25] Given \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and a weight \(w\), then \(w\in A_{p(\cdot)}\) if $$ \sup\limits_{\mathrm{B:ball}}|B|^{-1}\|w^{1/p(\cdot)}\chi_{B}\|_{L^{p(\cdot)}(w)}\|w^{-1/p(\cdot)}\chi_{B}\|_{L^{p'(\cdot)} (w)}< \infty .$$

Definition 1.6. [ 25] Given \(p_{1}(\cdot),~p_{2}(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\) and \(1/p_{1}(x)-1/p_{2}(x)=\mu/n\) such that \(0< \mu< n\). Then \(w\in A_{{p_{1}(\cdot)}, {p_{2}(\cdot)}}\) if $$ \|w\chi_{B}\|_{L^{p_{2}(\cdot)}}\|w^{-1}\chi_{B}\|_{L^{p'_{1}(\cdot)}}\leq|B|^{\frac{n-\mu}{n}} $$ holds for all balls \(B\in\mathbb{R}^{n}\).

Definition 1.7. [ 25] Let \(p(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\) and \(w\) be a weight. We say that \((p(\cdot),w)\) is an \(M\)-pair if the maximal operator \(M\) is bounded on \(L^{p(\cdot)}(w)\) and \(L^{p'(\cdot)}(w^{-1})\).

Now, we to need to give the definition of weighted Herz space with variable exponent. For all \(k \in{\mathbb{Z}}\), we denote \(B_{k}=\{ x\in\mathbb{R}^{n}: |x|\leq 2^{k}\},\) \(C_{k}= B_{k}\backslash B_{k-1},\) \(\chi_{k}= \chi_{C_{k}}.\)

Definition 1.8. [ 26] Suppose that \(p(\cdot)\in\mathcal{P}(\mathbb{R}^{n})\), \(0< q < \infty\), \(\alpha \in \mathbb{R}.\) The homogeneous weighted Herz space with variable exponent \(\dot{K}_{p(\cdot)}^{\alpha,q}(w)\) is the collection of \(f \in L_{\mathrm{loc}}^{p(\cdot)}(\mathbb{R}^{n} \backslash \{0\},w)\) such that $$ \|f\|_{\dot{K}^{\alpha,q}_{p(\cdot)}(w)} :=\left(\sum^{\infty}_{k=-\infty}2^{\alpha q k}\|f\chi_{k}\|^{q}_{L^{p(\cdot)}(w)}\right)^{1 /{q}}< \infty. $$ It is easy to see that if \(w = 1\), then \(\dot{K}_{p(\cdot)}^{\alpha,q}(w)= \dot{K}_{p(\cdot)}^{\alpha,q}(\mathbb{R}^{n})\) is the Herz space with variable exponen [ 17]. If \(w = 1\) and \(p(\cdot)=p\), then \( \dot{K}_{p(\cdot)}^{\alpha,q}(w)= \dot{K}_{p}^{\alpha,q}(\mathbb{R}^{n})\) is the classical Herz space introduced in [31]. If \(p(\cdot)=p\), then \( \dot{K}_{p(\cdot)}^{\alpha,q}(w)= \dot{K}_{p}^{\alpha,q}(w)\) is the weighted Herz space [ 32].

Definition 1.9. We say a kernel function \(\Omega(x , z )\) satisfies the \(L^{r}\)-Dini condition \((r\geq{1})\), if $$ \int_{0}^{1} \frac{\omega_{r}(\delta)}{\delta}(1+|\log\delta|^{\sigma})\mathrm{d}\delta < \infty, $$ where \({\omega_{r}(\delta)}\) denotes the integral modulus of continuity of order \(r\) of \(\Omega\) defined by $$ {\omega_{r}(\delta)}= \sup_{x\in{\mathbb{R}^{n}} , |\rho|< \delta } \left(\int\limits_{s^{n - 1}}|\Omega(x , \rho{z'}) - \Omega (x , z')|^{r}\mathrm{d}\sigma(z')\right)^{\frac{1}{r}}, $$ where \(\rho\) is the rotation in \(\mathbb{R}^{n}\), \(\|\rho\|=\sup\limits_{z'\in S^{n - 1}}\|\rho{z'}- {z'}\|.\)

Preliminaries and notations

In order to prove our main theorems, we need the following Lemmas.

Lemma 2.1.[3] Suppose that \(X\subset \mathcal{M}\) is a Banach function space.

1. (The generalized Hölder inequality) For all \(f\in X\) and \(g\in X'\), we have $$\int_{\mathbb{R}^{n}}|f(x)g(x)|\mathrm{d}x \leq \|f\|_{X}\|g\|_{X}.$$
2. For all \(f\in X\), we have $$ \sup\left\{\left|\int_{\mathbb{R}^{n}}f(x)g(x)\mathrm{d}x\right|:\|g\|_{X'} \leq 1 \right\}=\|f\|_{X}. $$

In particular, space \((X')'= X\).

As an application of the generalized Hölder inequality above, we have the following Lemma.

Lemma 2.2. Let \(X\) be a Banach function space, we have $$ 1\leq \frac{1}{|B|}\|\chi_{B}\|_{X}\|\chi_{B}\|_{X'} $$ hold for all balls \(B\).

Lemma 2.3.[24] Let \(X\) be a Banach function space. If the Hardy-Littlewood maximal operator \(M\) is weakly bounded on \(X\), that is $$ \|\chi_{\{Mf>\lambda\}}\|_{X}\leq \lambda^{-1}\|f\|_{X}, $$ holds for all \(f\in X\) and \(\lambda>0\)l, then we get $$ \sup\limits_{\mathrm{B: Ball}} \frac{1}{|B|}\|\chi_{B}\|_{X}\|\chi_{B}\|_{X'}< \infty. $$

Remark 2.1.[29] The weighted Banach function space \(X( \mathbb{R}^{n},W)\) is a Banach function space equipped the norm \(\|f\|_{X(\mathbb{R}^{n},W)}:=\|fW\|_{X}.\) The associated space of \(X( \mathbb{R}^{n},W)\) is a Banach function space and equals \(X'(\mathbb{R}^{n},W^{-1} )\).

Remark 2.2. If \(p_{1}(\cdot)\in \mathcal{P}(\mathbb{R}^{n})\), by comparing the definition of the weighted Banach function space with weighted variable Lebesgue space, we have

1. If \(X= L^{p_{1}(\cdot)}(\mathbb{R}^{n})\) and \(W = w\), then we obtain $$L^{p_{1}(\cdot)}(\mathbb{R}^{n}, w)= L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}).$$
2. If \(X= L^{p'_{1}(\cdot)}(\mathbb{R}^{n})\) and \(W = w^{-1}.\) Using Lemma 2.4, we obtain $$ L^{p'_{1}(\cdot)}(\mathbb{R}^{n}, w^{-1})= L^{p'_{1}(\cdot)}(w^{-p'_{1}(\cdot)})=(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'. $$

Lemma 2.4.[33] Suppose that \(X\) is a Banach space. Let \(M\) be bounded on the associated space \(X'\). Then there exists a constant \(0< \delta< 1 \) such that $$ \frac{\|\chi_{E}\|_{X}}{\|\chi_{B}\|_{X}}\leq \left(\frac{|E|}{|B|}\right)^{\delta} $$ holds for all balls \(B\) and all measurable sets \(E\subset B\).

Lemma 2.5.[26] Suppose that \(p_{1}(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\) and \(w^{p_{1}(\cdot)}\in A_{1}\). Let \(M\) be a bounded on \(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\) and \(L^{p'_{1}(\cdot)}(w^{-p'_{1}(\cdot)})\), then there exist constants \(\delta_{1},\delta_{2}\in (0,1) \) such that $$ \frac{\|\chi_{S}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}} {\|\chi_{B}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}} \leq C \left(\frac{|S|}{|B|}\right)^{\delta_{1}}, \frac{\|\chi_{S}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}} {\|\chi_{B}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}} \leq C \left(\frac{|S|}{|B|}\right)^{\delta_{2}}, $$ hold for all balls \(B\) and all measurable sets \(S\subset B\).

Lemma 2.6.[34] Suppose that \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}\) satisfies equation (1) and definition (1.9), \(\lambda>2\), \(2\rho -n>0\), \(1 < p < \infty\). Then for all \(f\in L^{q}(w)\) there exists \(C>0\) independent of \(f\) such that $$ \|\mu^{\rho}_{\Omega,s}f\|_{L^{q}(w)}\leq C \|f\|_{L^{q}(w)} $$ and $$ \|\mu^{*,\rho}_{\Omega,\lambda}f\|_{L^{q}(w)}\leq C \|f\|_{L^{q}(w)}. $$

Lemma 2.7.[25] Assume that for \(p_{0}\), \(1< p_{0}< \infty\) and every \(w_{0}\in A_{p_{0}}\), $$ \int\limits_{\mathbb{R}^{n}} f(x)^{p_{0}}w_{0}(x)\mathrm{d}x \leq \int\limits_{\mathbb{R}^{n}} g(x)^{p_{0}}w_{0}(x)\mathrm{d}x,\qquad\qquad\qquad(f,g)\in\mathcal{F}. $$ Then for any \(M\)-pair \((p(\cdot),w) \), $$ \|f\|_{L^{p(\cdot)}(w)}\leq C \|g\|_{L^{p(\cdot)}(w)},\qquad\qquad\qquad\qquad\qquad(f,g)\in\mathcal{F}, $$

Lemma 2.7 holds for \(p_{0}=1\) and the maximal operator is bounded on \(L^{p'(\cdot)}(w^{-1})\). We know the Hardy-Littlewood maximal operator is bounded on \(L^{p(\cdot)}(w)\) and \(L^{p'(\cdot)}(w^{-p'(\cdot)})\) (see [33]).
Combining Lemma 2.6 with Lemma 2.7, we obtain the following conclusion.

Corollary 2.8. Let \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\), \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{r}(S^{n-1})}(r\geq1)\) and \(w\in A_{p(\cdot)}\). Then the parameterized Littlewood-Paley operators \(\mu^{\rho}_{\Omega,s}\) and \(\mu^{*,\rho}_{\Omega,\lambda}\) with variable kernels are bounded on \(L^{p(\cdot)}(w)\).

3. Main Theorems and their proofs

In this section, we will prove the boundedness of the parameterized Littlewood-Paley operators with variable kernels on variable weighted Herz spaces.

Theorem 3.1. Let \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n})\), \(0< q_{1} \leq q_{2}< \infty\), \(\lambda>2\), \(2\rho -n>0\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}\) satisfies (1.1) and (1.2). If \(w^{p(\cdot)}\in A_{1}\) and \(-n\delta_{1}< \alpha < n \delta_{2} \), where \(\delta_{1},\delta_{2}\) are the constants in Lemma 2.5, then the operator \(\mu_{\Omega ,s}\) is bounded from \(\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\) to \(\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\).

Theorem 3.2. Let \(p(\cdot)\in LH(\mathbb{R}^{n})\cap \mathcal{P}(\mathbb{R}^{n}),\) \( 0< q_{1}\leq q_{2}< \infty\), \(\lambda>2\), \(2\rho -n>0\) and \(\Omega\in{L}^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}\) satisfies (1.1) and (1.2). If \(w^{p(\cdot)}\in A_{1}\) and \(-n\delta_{1}< \alpha< n\delta_{2}\), where \(\delta_{1},\delta_{2}\) are the constants in Lemma 2.5, then the operator \(\mu^{*}_{\Omega,\lambda}\) is bounded from \(\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\) to \(\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\).

Remark 3.1. As it is well known that, \(\mu^{\rho}_{\Omega,s}f(x)\leq 2^{n\lambda}\mu^{*,\rho}_{\Omega,\lambda}f(x)\) (see [6], p.89). Therefore, we give only the proof of Theorem 3.2.

Proof of Theorem 3.2 Let \(f\in\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})\). By the Jensen inequality, we have \begin{align*} \|\mu^{*,\rho}_{\Omega ,\lambda}f\|^{q_{1}}_{\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} &\leq \left(\sum^{\infty}_{k=-\infty}2^{\alpha q_{2} k}\|(\mu^{*,\rho}_{\Omega ,\lambda}f)\chi_{k}\|^{q_{2}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{\frac{q_{1}}{q_{2}}}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1} k}\|(\mu^{*,\rho}_{\Omega ,\lambda}f)\chi_{k}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} Denote \(f_{j}= f\chi_{j}\) for each \(j\in z\), then \(f = \sum_{j=-\infty}^{\infty}f_{j}\), so we have \begin{align*} \|\mu^{*,\rho}_{\Omega ,\lambda}f\|^{q_{1}}_{\dot{K}^{\alpha,q_{2}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\|(\mu^{*,\rho}_{\Omega ,\lambda}f)\chi_{k}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{k-2}_{j=-\infty}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{ k+2}_{j=k-2}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{\infty}_{j=k+2}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &=: \mathrm{L_{1}}+\mathrm{ L_{2}}+ \mathrm{L_{3}}. \end{align*} First, we consider \(\mathrm{L_{2}}\). Using Lemma 2.1 and \(-2\leq k-j\leq 2\), it is easy to get \begin{align*} \mathrm{L_{2}}&=\sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{ k+2}_{j=k-2}\|(\mu^{*,\rho}_{\Omega ,\lambda}f_{j})\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &=\sum^{\infty}_{k=-\infty}\left(\sum^{ k+2}_{j=k-2}2^{\alpha(k-j)}2^{\alpha j} \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\qquad\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}.\\ \end{align*} Now we need to consider \(\mu^{*,\rho}_{\Omega ,\lambda}f_{j}\). Applying the Minkowski inequality, we conclude that \begin{eqnarray*} &&|\mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)|=\\&&\left(\int^{\infty}_{0} \int_{\mathbb{R}^{n}}\left(\frac{t}{t + |x-y|}\right)^{\lambda n}\left|\frac{1}{t^{\rho}}\int_{|y-z|\leq t}\frac{\Omega(y,y-z)}{|y-z|^{n-\rho}}f_{j}(z)\mathrm{d}z\right|^{2}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\right)^{\frac{1}{2}}\\ &&\qquad\leq \int_{\mathbb{R}^{n}}f_{j}(z)\left(\int^{\infty}_{0}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\right)^{\frac{1}{2}} \mathrm{d}z\\ &&\qquad\leq \int_{\mathbb{R}^{n}}f_{j}(z)\left(\int^{|x-z|}_{0}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\right)^{\frac{1}{2}} \mathrm{d}z\\ &&\qquad+\int_{\mathbb{R}^{n}}f_{j}(z)\left(\int^{\infty}_{|x-z|}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\right)^{\frac{1}{2}} \mathrm{d}z. \end{eqnarray*} For \(\Omega\in{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}\) and \(2\rho-n>0\), the following inequality holds \begin{align*} \int_{|y-z|\leq t}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\mathrm{d}y &\leq\int_{s^{n-1}}\int^{t}_{0}\frac{|\Omega(sy'+z,y')|^{2}}{s^{2n-2\rho}}s^{n-1}\mathrm{d}s\mathrm{d}\sigma(y')\\ &\leq \|\Omega\|^{2}_{{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}} t^{2\rho-n}. \end{align*} Since \(|x-z|\leq|x-y|+|y-z|\leq|x-y|+t\). For \(\lambda > 2\), taking \(0< \delta < (\lambda -2)n \), we have \begin{eqnarray*} &&\int^{|x-z|}_{0}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\\ &&\leq {\int^{|x-z|}_{0} \int_{|y-z|\leq t}} \left(\frac{t}{t + |x-y|}\right)^{\lambda n-2n-\delta} \frac{1}{|x-z|^{2n+\delta}} \\&&\times\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho-n-\delta+1}}\\ &&\leq \frac{1}{|x-z|^{2n+\delta}} {\int^{|x-z|}_{0} \int_{|y-z|\leq t}}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho-n-\delta+1}}\\ &&\leq \frac{\|\Omega\|^{2}_{{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}}}{|x-z|^{2n+\delta}}\int^{|x-z|}_{0} t^{\delta-1}\mathrm{d}t\\ &&\leq C |x-z|^{-2n}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad(1.3) \end{eqnarray*} If we take \(1< \lambda_{1} < 2\), then \(\lambda_{1} n-n> 0\) and \(\lambda_{1} n-2n< 0\), so we have \begin{eqnarray*} &&\int^{\infty}_{|x-z|}\int_{|y-z|\leq t} \left(\frac{t}{t + |x-y|}\right)^{\lambda n}\frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho+n+1}}\\ &&\leq \int^{\infty}_{|x-z|}\int_{|y-z|\leq t} |x-z|^{-\lambda_{1} n} \frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-2\rho}}\frac{\mathrm{d}y\mathrm{d}t}{t^{2\rho-\lambda_{1}n+n+1}}\\ &&\leq \int^{\infty}_{|x-z|} |x-z|^{-\lambda_{1} n} \int_{|y-z|\leq t} \frac{|\Omega(y,y-z)|^{2}}{|y-z|^{2n-\lambda_{1} n}}\frac{\mathrm{d}y\mathrm{d}t}{t^{n+1}}\\ &&\leq \int^{\infty}_{|x-z|} |x-z|^{-\lambda_{1} n} \int_{s^{n-1}} \int^{t}_{0}\frac{|\Omega(y',(y-z)')|^{2}}{s^{2n-\lambda_{1}n}}s^{n-1}\mathrm{d}s\mathrm{d}\sigma(y') \frac{\mathrm{d}t}{t^{n+1}}\\ &&\leq C \|\Omega\|^{2}_{{L^{\infty}(\mathbb{R}^{n})\times{L^{2}(S^{n-1})}}} |x-z|^{-\lambda_{1} n} \int^{\infty}_{|x-z|} t^{\lambda_{1} n-2n-1}\mathrm{d}t\\ &&\leq C |x-z|^{-2n}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad(1.4) \end{eqnarray*} Combining the above two estimates, we obtain $$ \mu^{*}_{\Omega,\lambda}(f)(x)\leq \int_{\mathbb{R}^{n}}\frac{|f(z)|}{|x-z|^{n}} \mathrm{d}z. \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (1.5) $$ Next, we consider \(\mathrm{L_{1}}\). Noting that for \(x\in A_{k}\), \(z\in A_{j}\) and \(j\leq k-2\), then \(|x-z|\sim |x|\). By the virtue of the generalized Hölder's inequality, we have $$ \mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\leq C 2^{-kn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}. $$ Applying Lemma 2.3 and Lemma 2.5, we take \(\|.\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\) for each side, we have \begin{eqnarray*} &&\|\mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\chi_{k}\|_{{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}}\\ &&\leq C 2^{-kn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C 2^{-kn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{B_k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{B_k}\|^{-1}_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}\\ &&\leq C\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\frac{\|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}} {\|\chi_{B_k}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}}\\ &&\leq C 2^{(j-k)n\delta_{2}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{eqnarray*} Thus, we have \begin{align*} \mathrm{L_{1}} &\leq C \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{k-2}_{j=-\infty}2^{(j-k)n\delta_{2}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha } 2^{(k-j)(\alpha-n\delta_{2} )}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}. \end{align*} Now we have two cases: \(1< q_{1}< \infty\) and \(0< q_{1}\leq1\). When \(1< q_{1}< \infty\), by using the Hölder's inequality, we have \begin{align*} \mathrm{L_{1}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha } 2^{(k-j)(\alpha-n\delta_{2})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha q_{1} } 2^{(k-j)( \alpha-n\delta_{2} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right) \\&\times\left(\sum^{k-2}_{j=-\infty}2^{(k-j)(\alpha-n\delta_{2})q'_{1}/2}\right)^{q_{1}/q'_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{k-2}_{j=-\infty}2^{j\alpha q_{1} } 2^{(k-j)( \alpha-n\delta_{2} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{\infty}_{k=j+2} 2^{(k-j)( \alpha-n\delta_{2} )q_{1}/2}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} When \(0< q_{1}\leq1\), again by the Jensen inequality, we obtain \begin{align*} \mathrm{L_{1}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{k-2}_{j=-\infty}2^{j\alpha } 2^{(k-j)(\alpha-n\delta_{2})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{k-2}_{j=-\infty}2^{j\alpha q_{1} } 2^{(k-j)( \alpha-n\delta_{2} )q_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{\infty}_{k=j+2} 2^{(k-j)( \alpha-n\delta_{2} )q_{1}}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} Finally, we estimate \(\mathrm{L_{3}}\). Noting that for \(x\in A_{k}\), \(y\in A_{j}\) and \(j\geq k+2\), then \(|y-x|\sim |y|\). By (1.5) and the virtue of the generalized Hölder's inequality, we have $$ \mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\leq C 2^{-jn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'}. $$ Applying Lemma 2.3 and Lemma 2.5, we can take \(\|.\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\) for each side, we have \begin{eqnarray*} &&\|\mu^{*,\rho}_{\Omega,\lambda}(f_{j})(x)\chi_{k}\|_{{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}}\\ &&\leq C 2^{-jn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C 2^{-jn}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))'} \|\chi_{B_k}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &&\leq C \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\|\chi_{B_k}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))} \|\chi_{B_j}\|^{-1}_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))}\\ &&\leq C \|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\frac{\|\chi_{B_k}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))}} {\|\chi_{B_j}\|_{(L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)}))}}\\ &&\leq C 2^{(k-j)n\delta_{1}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{eqnarray*} Thus, we have \begin{align*} \mathrm{L_{3}} &\leq C \sum^{\infty}_{k=-\infty}2^{\alpha q_{1}k}\left(\sum^{\infty}_{j=k+2}2^{(k-j)n\delta_{1}}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha } 2^{(k-j)(\alpha+n\delta_{1} )}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}. \end{align*} Now we also have two cases: \(1< q_{1}< \infty\) and \(0< q_{1} \leq1\). When \(1< q_{1}< \infty\), by using the Hölder's inequality, we have \begin{align*} \mathrm{L_{3}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha } 2^{(k-j)(\alpha+n\delta_{1})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha q_{1} } 2^{(k-j)( \alpha+n\delta_{1} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)\\&\times \left(\sum^{\infty}_{j=k+2}2^{(k-j)(\alpha+n\delta_{1})q'_{1}/2}\right)^{q_{1}/q'_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{\infty}_{j=k+2}2^{j\alpha q_{1} } 2^{(k-j)( \alpha+n\delta_{1} )q_{1}/2}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{j-2}_{k=-\infty} 2^{(k-j)( \alpha+n\delta_{1} )q_{1}/2}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} When \(0< q_{1}\leq1\), applying the Jensen inequality, we obtain \begin{align*} \mathrm{L_{3}} &\leq C \sum^{\infty}_{k=-\infty}\left(\sum^{\infty}_{j=k+2}2^{j\alpha } 2^{(k-j)(\alpha+n\delta_{1})}\|f_{j}\|_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\right)^{q_{1}}\\ &\leq C \sum^{\infty}_{k=-\infty}\sum^{\infty}_{j=k+2}2^{j\alpha q_{1} } 2^{(k-j)( \alpha+n\delta_{1} )q_{1}}\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}\\ &\leq C \sum^{\infty}_{j=-\infty}2^{j\alpha q_{1} }\|f_{j}\|^{q_{1}}_{L^{p_{1}(\cdot)}(w^{p_{1}(\cdot)})} \sum^{j-2}_{k=-\infty} 2^{(k-j)( \alpha-n\delta_{2} )q_{1}}\\ &\leq C \|f\|^{q_{1}}_{\dot{K}^{\alpha,q_{1}}_{p_{1}(\cdot)}(w^{p_{1}(\cdot)})}. \end{align*} This completes the proof of Theorem 3.2.

Acknowledgments

The authors are very grateful to the referees for their valuable comments.

1. Introduction

Tablets and capsules represent unit dosage forms whereas liquid oral dosage forms such as syrups, suspensions, emulsions, solutions and elixirs usually contain one dose of medication in 5 to 30mL. Such doses are erratic by a factor ranging from 20 to 50% when the drug is self-administered by the patient [1]. The oral route of drug administration is the most important method for systemic effects. Solid oral dosage forms (tablet and capsule) are the preferred class of products of the two forms, the tablet has a number of advantages such as the tablet is an essentially tamper proof dosage form [2], [3]. Cetirizine is the carboxylated metabolite of hydroxyzine, and has high specific affinity for histamine H-receptors. Cetirizine dihydrochloride (CZ) is (RS)-2-[2-[4-[(4-chlorophenyl) phenyl methyl] piperazine-1-yl]ethoxy] acetic acid dihydrochloride. It is used for the symptomatic relief of allergic conditions including rhinitis and chronic urticarial [4], [5].

Literature survey shows comparative study of different brands of the same generic of pharmaceutical dosage form, and have indicated that pharmaceutical evaluation of different brands are as important as biological and clinical equivalency.

It is generally evaluated that many drugs that are manufactured in developing countries are implicated to be substandard [6]. To improve the quality of health products and to minimize the health risk factors. It is necessary to monitor all the pharmaceutical services in a regular basis that promoting the conditions and providing information on the base of which the people become enable to make healthy choices and they can make correct decisions about their health. Many tests are frequently applied to tablet dosage forms to render their optimum therapeutic effects. The technique of optimization is well reported in the literature for the development of tablet formulations [7, 8, 9]. The purpose of carrying out optimization is to select the best possible formulation from a pharmaceutical as well as consumer point of view.

In the context above we have made an attempt to compare all the physical and chemical parameters including purity of API's of five different brands of cetirizine 2HCl tablet available in local market of Lahore, Pakistan with that of multinational brand on the basis of GMP, GLP and Pharmcopial standards. Since no work has so far been carried out in the said perspective. Therefore we choosed this topic and to decide to provide awareness to allergy patients about most effective local brands with reasonable prices, without any health risk.

2. Materials and Methods

Reference cetirizine 2HCl was a kind gift sample from Jawa pharmaceuticals (pvt) Ltd. Lahore, Pakistan. Different brands of cetirizine 2HCl were obtained from different retail pharmacies of Lahore (Pakistan) and their qualitative and quantitative tests are carried out.
Zyrtec (UCB Farchim, S.A, Switzerland)
Alergo (Pharmix laboratories)
Cetrix (Saffron Pharmaceuticals)
Instazin (Friends Pharma )
Tryzin (Pulse pharmaceuticals)

3. Experimental

Spectrophotometer (Shimadzu UV-1800), Disintegration apparatus (IPL-27 model) Hot plate with magnetic stirrer (78-1) Hardness tester (PAK/HT-0607), Friability apparatus (Betatek EF-2), Weighing balance (Japan), micropipette, Hydrochloric acid (Merck Germany), and freshly prepared distilled water are used for the this study. Reasonable availability of medicine requires that a medicine should fulfill the clinical needs of a patient for an adequate period of time, at the lowest cost to them and their community. Various pharmaceutical parameters were employed as in USP 31 (2008), i.e. thickness, hardness, weight variation, friability, disintegration, and dissolution, to test the different brands of cetirizine hydrochloride tablets available.

3.1. Price Fluctuation

The most effective local brands with reasonable prices, without any health risk also provide awareness to the allergic patient in our community. Label information and price fluctuation among different brands of cetirizine 2HCl are given in Table 1.

Table 1. Label information of different brands of cetirizine 2HCl tablets (10mg)

Sr No	Product Name	Menufactured by	Batch No	Mfg date	Exp date	Price/10 tab
1	Zyrtec	UCB Farchim, S.A, Switzerland	HZXAP	04-2018	04-2020	42.00
2	Alergo	Pharmix laboratories (Pvt) Ltd	C824	03-2018	02-2021	39.00
3	Cetrix	Saffron Pharmaceuticals (Pvt)Ltd	106	11-16	10-19	28.00
4	Instazin	Friends Pharma (Pvt)Ltd	18IN021	08-2018	08-2021	42.00
5	Tryzin	Pulse pharmaceuticals (pvt)Ltd	1298	11-2018	10-2021	30.00

3.2. Thickness and Diameter

Thickness of tablet can vary without any change in its weight because of difference in the density of the granulation and the pressure applied to the tablets as well as speed of tablet compression. The thickness variation limits allowed are ± 5% of the size of the tablet [10, 11]. Vernier calipers were used to determine the thickness of 20 tablets. Table 2. shows mean ± S.D. of each brand tested.

3.3. Hardness Test

Hardness of the tablet is controlled by compression machine by applying the degree of pressure in Kg/cm2 during granules compression. This parameter is very important because it effects disintegration time and dissolution profile [12] A hardness tester (PAK/HT-0607)is used for 20 tablets randomly and mean standard deviation ± S. D. of each brand of cetirizine hydrochloride is calculated which shows in Table 2.

3.4. Weight Variation

The weight of the tablet dosage form is measured to check the proper amount of the active ingredient in the tablet. Analytical grade weighing balance is used to measure the individual as well as average weight of the tablet and mean standard deviation ± S.D. of each brand of cetirizine hydrochloride is calculated which shows in Table 2.

3.5. Disintegration Time

Disintegrations are required to break up tablets into fine powder in order to increase surface area of drugs exposed to gastrointestinal fluids. Disintegration test is conducted by disintegration apparatus (IPL-27) on different brands of cetirizine 2HCl 10mg tablets and it is found that all tablets of each brand disintegrated within the range of 04-22 minutes. Figure 3 indicates that product Instazin has shown a maximum average disintegration time about 22 minutes and sample Tryzin has shown minimum disintegration time about 2 minutes which is best as compare to other products. The disintegration test do serve as a component in the overall quality control of the tablets manufacturing. The results are shown in Table 3.

3.6. Friability Test

The friability test is designed to evaluate the ability of the tablet to withstand breakage during packaging, handling, and shipping. After physical inspection and dusting, ten tablets are weighed and placed in the friability apparatus (Betatek EF-2) where they are exposed to rolling and repeated shocks as they fall 6 inches in each tern within the apparatus. After 100 revolutions, the tablets are weighed and the weight is compared with the initial weight. The loss due to abrasion is a measure of tablet friability. Maximum weight loss is not more than 1% of the initial weight of the tablet. All the results are shown in Table 2.

3.7. Thin Layer Lhromatography (TLC)

Thin layer chromatography is used to separate, identify and purify of the compound or drug content. The \(R_{f}\) value of a particular pure analyte in a particular solvent (or mixture) is constant if the experimental conditions i.e. temperature, chromatography medium, solvent concentration and purity, amount of sample spotted on chromatography medium are kept unaltered. Thin layer chromatography is subject to check the specificity and purity of cetirizine hydrochloride in all brands by using solvent system ethyl acetate: methanol: formic acid (7.5:1.5:0.5 v/v/v) and compared it by Co-TLC with the standard. It is found that the active drug in all the brands showed single spot (\(R_{f}\) values 0.38) after visualization on TLC plate which confirmed the purity and specificity of the drug and it was also noted that no interference of any component on TLC was observed.

3.8. Preparation of Sample and Standard Solution

An accurately weighed 10mg cetirizine 2HCl (reference standards) is transferred to 100mL volumetric flask and dissolved in 0.1N HCl and make up the volume up to the mark with the same solvent to obtain standard solution having concentration 1000ppm. Magnetic stirrer is used for better dissolution. Further 10ppm dissolution is made by taking 1mL from the above solution and make up the volume to 100mL with 0.1N HCl.

The standard solution of cetirizine 2HCl is made by dissolving 10mg of pure cetirizine 2HCl in 100mL of 0.1N HCl. Magnetic stirrer is used for better dissolution. Further 10ppm dissolution is made by taking 1mL from the above solution and make up the volume to 100mL with 0.1N HCl.

3.9. Content assay

Every tablet contains the amount of active ingredient claimed, which can vary among the tablets within a batch. Analysis of drug potency in tablet not only indicating the presence of drug in dosage form but also shows its stability. For the evaluation of content, assay has been performed. The chemical assay results showed that all the five brands have active ingredient amount within limits i.e 90-110%.

Table 2. Physiochemical Parameter of cetirizine 2HCl Tablets

Brand names	Thickness (mm)	Diameter (mm)	Hardness test (kg/cm2)	Weight variation (mg)	Friability test (%age)
Zyrtec	2.71mm	4.10mm	3.08 kg/cm2	118 mg	NMT 1%
Alergo	3.51mm	5.55mm	3.46 kg/cm2	150 mg	NMT 1%
Cetrix	3.08mm	7.99mm	2.39 kg/cm2	160 mg	NMT 1%
Instazin	2.90mm	5.89mm	3.27 kg/cm2	150 mg	NMT 1%
Tryzin	3.11mm	4.03mm	3.13 kg/cm2	140 mg	NMT 1%

Table 3. Physiochemical Parameter of cetirizine 2HCl Tablets

Brand names	Disintegration time (minutes)	\(R_{f}\) values	Dissolution test (%age)	Chemical assay (%age)
Zyrtec	4 minutes	0.38	89.65%	103%
Alergo	5 minutes	0.38	85.74 %	102%
Cetrix	5 minutes	0.38	87.95%	106%
Instazin	22 minutes	0.38	84.25%	103%
Tryzin	2 minutes	0.38	82.70%	105%

3.10. Dissolution Studies

Dissolution studies are conducted using a USP apparatus II, paddle type with 50 rpm at 37°±1C. For standard preparation, about 10 mg of cetirizine hydrochloride is placed in a 100mL volumetric flask and dissolved with 0.1 M hydrochloric acid and then the volume was made up to 100 mL with 0.1 M hydrochloric acid. 2 mL of this solution is transferred to another 100mL volumetric flask and diluted to 100mL with the same solvent. For the sample, about 900 mL of 0.1 mL HCl was placed in the dissolution bowl with one tablet and the apparatus was started. The sample is drawn at time intervals of 5, 10, 15, 30 and 45 minutes for each formulation. Absorbance of the sample preparation and that of standard were taken at 220nm using a 0.1M hydrochloric acid solution as a blank. Drug concentrations are measured spectrophotometrically. Results are shown in Table 3.

4. Results and Discussion

Price fluctuation among different brands of cetirizine 2HCl is given in Figure 1. Weight variation comparison of different brands of cetirizine 2HCl is given in Figure 2. Comparison of disintegration time of all the five samples is shown in Figure 3 and comparison of % Assay of all the five samples is shown in Figure 4. It is evaluated by literatures that many drugs which are manufactured in developing countries are implicated to be substandard. In order to eliminate the health risk and provide the local community with affordable quality drugs, it is necessary to check the pharmaceutical services at regular basis. By providing this sort of information, people can make healthy choices according to their health problems. Relative to the quality of drug, the price fluctuation is also noticed in the societies where there is no regulatory control. In Figure. 1 the graphical representation shows that a significant variation in the cost of different brands containing same salt i.e. cetirizine 2HCl is present, whereas there is not a significant variation present in their quality. So its use can reduce the health expenses of a patient.

A drug becomes publically famous, by its effectiveness, immediate results and lesser side effects. The effectiveness in turn depends upon the content of active ingredients, disintegration time of drug, and its dissolution. All these parameters are controlled and monitored by Pharmaceutical quality control and quality assurance department. These departments check the composition and uniformity of the drug substances used in processing and in the final product. In case of above mentioned drugs from local pharmaceuticals, the traditional tests have been used to compare them with that of a multinational company. For this purpose, we conducted their physicochemical parameter's evaluation, to get a stable and effective drug product which may be pharmaceutically equivalent to a multinational product. The safety and efficiency of these drugs was measured by uniformity content. The amount of cetirizine 2HCl in the different local brands available in Lahore, is proved to be within limits, so these can replace a multinational brand at a low cost.

The present study is based on spectrophotometric method for chemical assay, TLC technique for the purity of API's and physiochemical parameters i.e. weight variation, hardness, friability, thickness and disintegration time as well as price fluctuation in PK rupees. Weight variation of the tablets were examined; all the tablets weight are in accordance with the required limit (NMT ±5%), hardness is tested; the hardness is in good agreement with the specification (5-10 kg/cm2), thickness and the friability of the tablets are determined which complied with the specification (3.5 mm and NMT 1% ) respectively; disintegration test is performed, all the tablets were disintegrated with the prescribed time ( NMT 30 minutes); TLC technique is developed to check the quality and purity of API, all the active ingredients showed the similar \(R_{f}\)-values without any impurity (0.38) ; pharmaceutical assay was carried out, none had potency less than the required specification ( 90% to 110% ).

From the comparative study, it is observed that all the brands of cetirizine 2HCl, tablets (10mg) showed the same results as per I.P standards .Since there is a variation in the formulation of these drugs which may affect the results by performing qualitative and quantitative tests. From the test results and plots, it is observed and concluded that an economical and quality products can be prescribed for allergy patients whether they are manufactured by the local or multinational pharmaceutical companies without any health risk.

Competing Interests

The authors declare that they have no competing interests.