In this article, we are mainly interested to find some sufficient conditions for integral operator involving normalized Struve and Dini function to be in the class \(N\left( \mu \right)\). Some corollaries involving special functions are also the part of our investigations.
Lemma 2. 1. If \(b,v \in \mathbb{R}\), and \(c\in \mathbb{C}\), \(k=v+\frac{b+2}{2}\) are so constrained that \begin{equation*} k>\max \left\{ 0,\frac{7\left\vert c\right\vert }{24}\right\} , \end{equation*} then the function \(u_{v,b,c}:\mathcal{U}\rightarrow \mathbb{C}\) defined by (8) satisfies the following inequality: \begin{equation*} \left\vert \frac{zu_{v,b,c}^{\prime }(z)}{u_{v,b,c}(z)}-1\right\vert \leq \frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{% 3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\ \ \ \ \left( z\in \mathcal{U}\right) . \end{equation*}
Lemma 2.2 . Let \(v\in \mathbb{R}\) and consider the normalized Dini function \(q_{v}(z):\mathcal{U}\rightarrow \mathbb{C},\) defined by \begin{equation*} q_{v}\left( z\right) =2^{v-1}\Gamma \left( v+1\right) z^{1-\frac{v}{2}% }\left( \left( 2-v\right) J_{v}\left( \sqrt{z}\right) +\sqrt{z}J_{v}^{\prime }\left( \sqrt{z}\right) \right) , \end{equation*} where \(J_{v}(z)\) is the Bessel function of first kind. Then \begin{equation*} \left\vert \frac{zq_{v}^{\prime }(z)}{q_{v}(z)}-1\right\vert \leq \frac{4v+9 }{2\left( 4v^{2}+9v+3\right) },\text{ \ }v>\frac{-9+\sqrt{33}}{8}. \end{equation*} The main objective of this paper is to give sufficient conditions for integral operator involving some special functions. The main results are given below.
Theorem 3.1 . Let \(v_{1},\ldots ,v_{n},\) \(b\in \mathbb{R}\), \(c\in\mathbb{C}\) and \(k_{i}>\frac{7\left\vert c\right\vert }{24}\) with \(k_{i}=v_{i}+\left( b+2\right) /2,\ i=1,\ldots ,n.\) Let \(u_{v_{i},b,c}:\mathcal{U}\rightarrow\mathbb{C}\) be defined as \begin{equation*} u_{v_{i},b,c}(z)=2^{v}\sqrt{\pi }\Gamma \left( v+\frac{b+2}{2}\right) z^{% \frac{\left( -1-v\right) }{2}}w_{v_{i},b,c}(\sqrt{z}). \end{equation*} Suppose \(k=\min \left\{ k_{1},k_{2},\ldots ,k_{n}\right\} \), \(\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}\) \((i=1,2,…,n)\) are positive real numbers and let \(f_{j}(z),\) \((j=1,2,…,m)\) be of the form of (1) is in class \(N\left( p,\gamma _{j}\right)\). More over\ these numbers satisfy the relation \begin{equation*} 1<1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) <\frac{% 2^{p}+1}{2^{p-1}+1}, \end{equation*} then the function \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}:\mathcal{U}\rightarrow \mathbb{C}\) defined by (12) is in \(N\left( \mu \right) \), where \begin{equation*} \mu =1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*}
Proof. We easily observe that \(u_{v_{i},b,c},\forall i=1,2,\cdots n\) are analytic and normalized of the form of \(u_{v_{i},b,c}\left( 0\right) =u_{v_{i},b,c}^{\prime }\left( 0\right) -1=0.\) Clearly \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\) also analytic and normalized form of the form of \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\left( 0\right) =F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( 0\right) -1=0.\) On the other hand, it is easy to see that \begin{equation*} F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\left( z\right) =\overset{n}{\underset{i=1}{\prod }}\left( \frac{% u_{v_{i},b,c}}{z}\right) ^{\alpha _{i}}\overset{n}{\underset{i=1}{\prod }}% \left( \frac{D^{p}f_{j}(z)}{z}\right) ^{\beta _{j}}. \end{equation*} Differentiating logarithmically, we get \begin{eqnarray*} &&\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{% zu_{v_{i},b,c}^{\prime }(z)}{u_{v_{i},b,c}\left( z\right) }-1\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}% -1\right) , \end{eqnarray*} or equivalently, \begin{eqnarray*} &&1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) } \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{zu_{v_{i},b,c}^{\prime }(z)}{% u_{v_{i},b,c}\left( z\right) }\right) +\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}\right)\\ &+&1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}. \end{eqnarray*} This implies that \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}Re\left( \frac{zu_{v_{i},b,c}^{% \prime }(z)}{u_{v_{i},b,c}\left( z\right) }\right) +\sum\limits_{j=1}^{m}\beta _{j}Re\left( \frac{D^{p+1}f_{j}(z)}{% D^{p}f_{j}(z)}\right)\\ &+&\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) . \end{eqnarray*} Now, by using Lemma 2.1 for each \(v_{i}\), where \(i=1,2,\cdots n,\) we obtain \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &\leq &\sum\limits_{i=1}^{n}\alpha _{i}\left( 1+\frac{\left\vert c\right\vert \left( 6k_{i}-\left\vert c\right\vert \right) }{3\left( 4k_{i}-\left\vert c\right\vert \right) \left( 3k_{i}-\left\vert c\right\vert \right) }\right) \\ &+& \sum\limits_{j=1}^{m}\beta _{j}\gamma _{j}+\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) \\ &=&1+\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{\left\vert c\right\vert \left( 6k_{i}-\left\vert c\right\vert \right) }{3\left( 4k_{i}-\left\vert c\right\vert \right) \left( 3k_{i}-\left\vert c\right\vert \right) }\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Now consider the funnction \begin{equation*} \tau :\left( \frac{7\left\vert c\right\vert }{24},\infty \right) \rightarrow\mathbb{R}, \end{equation*} defined by \begin{equation*} \tau \left( k\right) =\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }, \end{equation*} is decreasing function \begin{equation*} \frac{\left\vert c\right\vert \left( 6k_{i}-\left\vert c\right\vert \right) }{3\left( 4k_{i}-\left\vert c\right\vert \right) \left( 3k_{i}-\left\vert c\right\vert \right) }\leq \frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }. \end{equation*} Therefore \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &\leq& 1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Since \(1<1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) <\frac{% 2^{p}+1}{2^{p-1}+1},\) therefore \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\in N\left( \mu \right) \), where \begin{equation*} \mu =1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*} which completes the proof.
By setting \(\beta _{1}=\beta _{2}=…=\beta _{n}=0\) and \(p=1\) in Theorem 3.1 we will obtain the result given belowCorollary 3.2. Let \(v_{1},\ldots ,v_{n},\) \(b\in\mathbb{R}\), \(c\in\mathbb{C}\) and \(k_{i}>\frac{7\left\vert c\right\vert }{24}\) with \(k_{i}=v_{i}+\left( b+2\right) /2,\ i=1,\ldots ,n.\) Let \(u_{v_{i},b,c}:\mathcal{U}\rightarrow\mathbb{C}\) be defined as \begin{equation*} u_{v_{i},b,c}(z)=2^{v}\sqrt{\pi }\Gamma \left( v+\frac{b+2}{2}\right) z^{% \frac{\left( -1-v\right) }{2}}w_{v_{i},b,c}(\sqrt{z}). \end{equation*} Suppose \(v=\min \left\{ v_{1},v_{2},…,v_{n}\right\}\). Let \(\alpha _{1},\ldots ,\alpha _{n}\) \((i=1,2,…,n)\) are positive real numbers and \(f_{j}(z),\) \((j=1,2,…,m)\) be of the form of (1) is in class \(N\left( p,\gamma _{j}\right)\). More over these numbers satisfy the relation
\begin{equation*} 1<1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}<\frac{3}{2}, \end{equation*} then the function \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}:\mathcal{U}\rightarrow\mathbb{C}\) defined by (12) is in \(N\left( \mu \right) ,\) where \begin{equation*} \mu =1+\frac{\left\vert c\right\vert \left( 6k-\left\vert c\right\vert \right) }{3\left( 4k-\left\vert c\right\vert \right) \left( 3k-\left\vert c\right\vert \right) }\sum\limits_{i=1}^{n}\alpha _{i}. \end{equation*} The next theorem gives other sufficient conditions for the integral operator defined in (12). The key tool in the proof is Lemma 2.2.Theorem 4.1 . Let \(v_{1},\ldots ,v_{n}>\frac{-9+\sqrt{33}}{8},\) where \(n\in \mathbb{N}\). Let \(q_{v_{i}}:\mathcal{U}\rightarrow \mathbb{C}\) be defined as \begin{equation*} q_{v_{i}}\left( z\right) =2^{v_{i}-1}\Gamma \left( v_{i}+1\right) z^{1-\frac{% v_{i}}{2}}\left( \left( 2-v_{i}\right) J_{v_{i}}\left( \sqrt{z}\right) + \sqrt{z}J_{v_{i}}^{\prime }\left( \sqrt{z}\right) \right) . \end{equation*} Suppose \(k=\min \left\{ k_{1},k_{2},\ldots ,k_{n}\right\} \), \(\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}\) are positive real numbers and let \(f_{j}(z),\) \((j=1,2,…,m)\) be of the form of (1) is in class \(N\left( p,\gamma _{j}\right) .\) More over these numbers satisfy the relation \begin{equation} 1<1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) <\frac{% 2^{p}+1}{2^{p-1}+1}, \label{m16} \end{equation} then the function \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}:\mathcal{U}\rightarrow \mathbb{C}\) defined by (12) is in \(N\left( \mu \right) ,\) where \begin{equation*} \mu =1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*}
Proof. We easily observe that \(q_{v_{i}}\forall i=1,2,\cdots n\) are analytic and normalized of the form of \(q_{v_{i}}\left( 0\right) =q_{v_{i}}^{\prime }\left( 0\right) -1=0.\) Clearly \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\) also analytic and normalized form of the form of $$F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\left( 0\right) =F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( 0\right) -1=0.$$ On the other hand, it is easy to see that \begin{equation*} F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\left( z\right) =\overset{n}{\underset{i=1}{\prod }}\left( \frac{ q_{v_{i}}}{z}\right) ^{\alpha _{i}}\overset{n}{\underset{i=1}{\prod }}\left( \frac{D^{p}f_{j}(z)}{z}\right) ^{\beta _{j}}. \end{equation*} Differentiating logarithmically, we get \begin{eqnarray*} &&\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{ zq_{v_{i}}^{\prime }(z)}{q_{v_{i}}\left( z\right) }-1\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)} -1\right) , \end{eqnarray*} or equivalently, \begin{eqnarray*} &&1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) } \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{q_{v_{i}}^{\prime }(z)}{% q_{v_{i}}\left( z\right) }\right) +\sum\limits_{j=1}^{m}\beta _{j}\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}\right) +1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}. \end{eqnarray*} This implies that \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{ F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &=&\sum\limits_{i=1}^{n}\alpha _{i}Re\left( \frac{zq_{v_{i}}^{\prime }(z)}{q_{v_{i}}\left( z\right) }\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j} Re\left( \frac{D^{p+1}f_{j}(z)}{D^{p}f_{j}(z)}\right) +\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) . \end{eqnarray*} Now, by using Lemma 2.2 for each \(v_{i}\), where \(i=1,2,\cdots n,\) we obtain \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{ F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\right\} \\ &\leq &\sum\limits_{i=1}^{n}\alpha _{i}\left( 1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\right)\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\gamma _{j}+\left( 1-\sum\limits_{i=1}^{n}\alpha _{i}-\sum\limits_{i=1}^{n}\beta _{j}\right) \\ &=&1+\sum\limits_{i=1}^{n}\alpha _{i}\left( \frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\right)\\ &+& \sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Now as it is clear that the function \begin{equation*} \phi (v):\left( \frac{-9+\sqrt{33}}{8},\infty \right) \rightarrow \mathbb{R}, \end{equation*} defined by \begin{equation*} \phi (v)=\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }, \end{equation*} is decreasing function. Therefore \begin{equation*} \frac{4v_{i}+9}{2\left( 4v_{i}^{2}+9v_{i}+3\right) }\leq \frac{4v+9}{2\left( 4v^{2}+9v+3\right) }. \end{equation*} Therefore \begin{eqnarray*} &&Re\left\{ 1+\frac{zF_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime \prime }\left( z\right) }{% F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}^{\prime }\left( z\right) }\right\}\\ &\leq&1+\frac{4v+9}{2\left(4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha_{i}\\ &+&\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{eqnarray*} Since \(1<1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n} \alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) < \frac{2^{p}+1}{2^{p-1}+1},\) therefore $$F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}\in N\left( \mu \right) $$ , where \begin{equation*} \mu =1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}+\sum\limits_{j=1}^{m}\beta _{j}\left( \gamma _{j}-1\right) . \end{equation*} which completes the proof.
By setting \(\beta _{1}=\beta _{2}=…=\beta _{n}=0\) and \(p=1\) in Theorem 4.1 we will obtain the result given belowCorollary 4.2. Let \(v_{1},\ldots ,v_{n}>\frac{-9+\sqrt{33}}{8},\) where \(n\in\mathbb{N}\). Let \(q_{v_{i}}:\mathcal{U}\rightarrow \mathbb{C}\) be defined as \begin{equation*} q_{v_{i}}\left( z\right) =2^{v_{i}-1}\Gamma \left( v_{i}+1\right) z^{1-\frac{% v_{i}}{2}}\left( \left( 2-v_{i}\right) J_{v_{i}}\left( \sqrt{z}\right) +% \sqrt{z}J_{v_{i}}^{\prime }\left( \sqrt{z}\right) \right) . \end{equation*} Suppose \(v=\min \left\{ v_{1},v_{2},…,v_{n}\right\}\). Let \(\alpha _{1},\ldots ,\alpha _{n}\) \((i=1,2,…,n)\) are positive real numbers and \(f_{j}(z),\) \((j=1,2,…,m)\) be of the form of (1) is in class \(N\left( p,\gamma _{j}\right).\) More over these numbers satisfy the relation \begin{equation*} 1<1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}<\frac{3}{2}, \end{equation*} then the function \(F_{v_{1},\ldots ,v_{n},\alpha _{1},\ldots ,\alpha _{n},\beta _{1},…,\beta _{m}}:\mathcal{U}\rightarrow \mathbb{C}\) defined by (12) is in \(N\left( \mu \right)\), where \begin{equation*} \mu =1+\frac{4v+9}{2\left( 4v^{2}+9v+3\right) }\sum\limits_{i=1}^{n}\alpha _{i}. \end{equation*}