<!DOCTYPE html>
<html lang="en">
<head>
<script>
  MathJax = { 
    tex: {
		    inlineMath: [['\\(','\\)']]
	} }
</script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
<meta name="generator" content="plasTeX" />
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Certain Properties for a Class of Analytic Functions Associated with Hypergeometric Functions : Certain Properties for a Class of Analytic Functions Associated with Hypergeometric Functions </title>
<link rel="stylesheet" href="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/styles/theme-white.css" />
</head>

<body>

<div class="wrapper">

<div class="content">
<div class="content-wrapper">


<div class="main-text">


<div class="titlepage">
<h1>Certain Properties for a Class of Analytic Functions Associated with Hypergeometric Functions </h1>
<p class="authors">
<span class="author">Khadeejah Rasheed Alhindi\(^\ast \) Maslina Darus\(^\S \)</span>
</p>
<p class="date">February 24, 2014.<br />Published online: January 23, 2015.</p>
</div>
<p>\(^\ast \)School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia, e-mail: <span class="tt">m.kh83@yahoo.co.uk</span>. The work of this author has been supported by Zamalah and GUP-2013-004 </p>
<p>\(^\S \)School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia, e-mail: <span class="tt">maslina@ukm.edu.my</span> (Corresponding author). The work of this author has been supported by GUP-2013-004 </p>

<div class="abstract"><p> In this particular paper, we investigate coefficient inequalities, closure theorems, convolution properties for the functions belonging to the class \( \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). Further, integral transforms of functions in the same class are also discussed. </p>
<p><b class="bf">MSC.</b> 30C45, 33D15, 05A30 </p>
<p><b class="bf">Keywords.</b> Analytic function, coefficient inequalities, Schwarz inequality, closure theorems, Hadamard product, convolution properties, integral operator. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(\mathcal{A}\) be the class of analytic functions of the form </p>
<div class="equation" id="one">
<p>
  <div class="equation_content">
    \begin{equation} \label{one} f(z)=z+\sum _{k=2}^{\infty } a_k z^k;\mathrm\qquad z\in (\mathbb {U}=\{ z\in {C}:|z|<1\} ) \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p> and \(\mathcal{S}\) be the subclass of \(\mathcal{A}\) consisting of <i class="itshape">univalent functions</i>, and \(\mathcal{S(\alpha )}\), \(\mathcal{C(\alpha )}\) \((0 {\lt} \alpha \leq 1)\) denote the subclasses of \(\mathcal{A}\) consisting of functions that are starlike of order \(\alpha \) and convex of order \(\alpha \) in \(\mathbb {U}\), respectively. </p>
<p>For two analytic functions \( f(z)=z+\sum _{k=2}^{\infty } a_k z^k \) and \( g(z)=z+\sum _{k=2}^{\infty } b_k z^k\) in the open unit disc \(\mathbb {U}=\{ z\in {C}:|z|{\lt}1\} \), the Hadamard product (or convolution) \(f\ast g\) of \(f\) and \(g\) is defined by </p>
<div class="equation" id="hadamard">
<p>
  <div class="equation_content">
    \begin{equation} \label{hadamard} f(z)\ast g(z) = (f*g)(z)=z+\sum _{k=2}^{\infty } a_k b_k z^k. \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p> For complex parameters \(\alpha _1,...\alpha _r\) and \(\beta _1,...\beta _s\) \((\beta _j\neq 0,-1,-2,...;j=1...s)\), Dziok and Srivastava <span class="cite">
	[
	<a href="#dziok" >1</a>
	]
</span> defined the generalized hypergeometric function </p>
<div class="displaymath" id="a0000000003">
  \[ _rF_s(\alpha _1,...,\alpha _r;\beta _1,...,\beta _s;z) \]
</div>
<p> by </p>
<div class="equation" id="F">
<p>
  <div class="equation_content">
    \begin{equation} \label{F} _rF_s(\alpha _1,...,\alpha _r;\beta _1,...,\beta _s;z)=\sum _{k=0}^{\infty }\tfrac {(\alpha _1)_k...(\alpha _r)_k}{(\beta _1)_k,...,(\beta _s)_k }\tfrac {z^k}{k!}; \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<div class="equation" id="a0000000004">
<p>
  <div class="equation_content">
    \begin{equation}  (r\leq s+1; r,s \in \mathbb {N}_0= \mathbb {N} \cup {0}; z \in \mathcal{U}), \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p> where \((x)_k\) is the Pochhammer symbol defined, in terms of Gamma function \(\Gamma \), by </p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  (x)_k =\tfrac {\Gamma (x+k)}{\Gamma (x)}= \begin{cases}  1, &  \text{if } k=0,\\ x(x+1)...(x+k-1), &  \text{if } k\in \mathbb {N}. \end{cases}\end{equation*}
</div>
<p>Dziok and Srivastava <span class="cite">
	[
	<a href="#dziok" >1</a>
	]
</span> defined also the linear operator </p>
<div class="equation" id="H">
<p>
  <div class="equation_content">
    \begin{equation} \label{H} H(\alpha _1,...,\alpha _r;\beta _1,...,\beta _s)f(z)= z + \sum _{k=2}^{\infty } \Gamma _k a_k z^k \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p> where </p>
<div class="equation" id="g">
<p>
  <div class="equation_content">
    \begin{equation} \label{g} \Gamma _k = \frac{(\alpha _1)_{k-1}...(\alpha _r)_{k-1}}{(\beta _1)_{k-1},...,(\beta _s)_{k-1}(k-1)!}. \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>Al-Abbadi and Darus <span class="cite">
	[
	<a href="#abbadi" >2</a>
	]
</span> defined the analytic function </p>
<div class="equation" id="a0000000006">
<p>
  <div class="equation_content">
    \begin{equation}  \Phi ^m_{\lambda _1,\lambda _2}=z+\sum _{k=2}^{\infty }\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}z^k, \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p> where \(m\in \mathbb {N}_0 = \{ 0,1,2,....\} \) and \( \lambda _2 \geq \lambda _1 \geq 0\). </p>
<p>Using the Hadamard product (<a href="#hadamard">2</a>), we can derive the generalized derivative operator \(\mathcal{K}^{m,r,s}_{\lambda _1,\lambda _2} \) as follows </p>
<div class="equation" id="a0000000007">
<p>
  <div class="equation_content">
    \begin{equation}  \mathcal{K}^{m,r,s}_{\lambda _1,\lambda _2}f(z)=z+\sum _{k=2}^{\infty }\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m} {\Gamma _k} a_k z^k \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p> where \( \Gamma _k\) is as given in (<a href="#g">8</a>). <div class="remark_thmwrapper " id="a0000000008">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1</span>
  </div>
  <div class="remark_thmcontent">
  <p>When (\(\lambda _1=\lambda _2=0\)), (\(\lambda _1=m=0\)) or (\(\lambda _2=0\) and \(m=1\)) we get Dziok-Srivastava operator <span class="cite">
	[
	<a href="#dziok" >1</a>
	]
</span>. </p>
<p>Also there are three cases to get the Hohlov operator <span class="cite">
	[
	<a href="#hohlov" >3</a>
	]
</span>, by giving (\(\lambda _1=\lambda _2=0,\alpha _i=0,\beta _j=0 \)), (\(\lambda _1=m=0,\alpha _i=0,\beta _j=0\)) or (\(\lambda _2=0, m=1, \alpha _i=0, \beta _j=0\)) where (\(i=1...r \) and \(j=1...s\)). </p>
<p>Putting (\(\lambda _1=\lambda _2=0, \alpha _2=1,\alpha _3=...=\alpha _r=0,\beta _2=...=\beta _s=0\)), (\(\lambda _1=m=0, \alpha _2=1,\alpha _3=...=\alpha _r=0,\beta _2=...=\beta _s=0\)) or \((\lambda _2=0, m=1, \alpha _2=1,\alpha _3=...=\alpha _r=0,\beta _2=...=\beta _s=0)\), we obtain the Carlson-Shaffer operator <span class="cite">
	[
	<a href="#carlson" >4</a>
	]
</span>. </p>
<p>There are six cases to get the Ruscheweyh operator <span class="cite">
	[
	<a href="#ruscheweyh" >5</a>
	]
</span> as follows: (\(\lambda _1=\lambda _2=0,\alpha _2=\alpha _3=...=\alpha _r=0,\beta _1=\beta _2=...=\beta _s=0\)), (\(\lambda _1=m=0,\alpha _2=\alpha _3=...=\alpha _r=0,\beta _1=\beta _2=...=\beta _s=0\)), (\(\lambda _2=0, m=1,\alpha _2=\alpha _3=...=\alpha _r=0,\beta _1=\beta _2=...=\beta _s=0\)),(\(\lambda _1=\lambda _2=0, \alpha _2=\alpha _3=...=\alpha _r=0,\beta _2=...=\beta _s=0\)), (\(\lambda _1=m=0, \alpha _2=\alpha _3=...=\alpha _r=0,\beta _2=...=\beta _s=0\)) or \((\lambda _2=0, m=1, \alpha _2=\alpha _3=...=\alpha _r=0,\beta _2=...=\beta _s=0)\). </p>
<p>If(\(\lambda _2=0,m=2,\alpha _2=\alpha _3=...=\alpha _r=0,\beta _1=\beta _2=...=\beta _s=0\)), we get the generalized Ruscheweyh derivative operator as well <span class="cite">
	[
	<a href="#shaqsi" >6</a>
	]
</span> . </p>
<p>Moreover, if we put (\(\alpha _2=\alpha _3=...=\alpha _r=0,\beta _1=\beta _2=...=\beta _s=0\)) or (\(\alpha _2=\alpha _3=...=\alpha _r=0,\beta _1=\beta _2=...=\beta _s=0\)), we can get operator given by Al-Abbadi and Darus <span class="cite">
	[
	<a href="#abbadi" >2</a>
	]
</span>. </p>
<p>After that, if(\(\lambda _2=0, m=m+1, \alpha _2=\alpha _3=...=\alpha _r=0,\beta _1=\beta _2=...=\beta _s=0\)), we get the generalized derivative operator by Al-Shaqsi and Darus <span class="cite">
	[
	<a href="#darus" >7</a>
	]
</span>. <br /><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000009">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(f\in \mathcal{A}\). Then \(f(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) if and only if </p>
<div class="equation" id="a0000000010">
<p>
  <div class="equation_content">
    \begin{equation}  \Re \bigg\{ \frac{z[\mathcal{K}^{m,r,s}_{\lambda _1,\lambda _2}f(z)]'}{\mathcal{K}^{m,r,s}_{\lambda _1,\lambda _2}f(z)}\bigg\} >\eta ,\quad 0\leq \eta <1, z\in \mathcal{U}. \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>

  </div>
</div> </p>
<p>In this present paper, we obtain the coefficient inequalities, closure theorems, convolution properties for the functions belonging to the class \( \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). Finally, the preserving integral operators of the form </p>
<div class="equation" id="a0000000011">
<p>
  <div class="equation_content">
    \begin{equation}  G_c(z)=c\int _0^1u^{c-2}f(uz)du; \quad \quad (c>0) \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p> for the class \(\mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) is considered. We employ techniques similar to those used earlier by <span class="cite">
	[
	<a href="#owa" >8</a>
	]
</span>. </p>
<h1 id="a0000000012">2 Coefficient Estimate for the Class \(\mathcal{S}^{\lowercase {m,r,s}}_{\lambda _1,\lambda _2}(\eta )\) </h1>
<p> <div class="theorem_thmwrapper " id="coef">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let \(f(z)\in \mathcal{A}\). If </p>
<div class="equation" id="co">
<p>
  <div class="equation_content">
    \begin{equation} \label{co} \sum _{k=2}^\infty (k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} |a_k|\leq 1-\eta ,\qquad 0\leq \eta \leq 1 \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p> then \(f(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). The result <span class="rm">(<a href="#co">13</a>)</span> is sharp. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000013">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Suppose that (<a href="#co">13</a>) holds. Since </p>
<div class="displaymath" id="a0000000014">
  \begin{eqnarray*}  1-\eta &  \geq &  \sum _{k=2}^\infty (k-\eta ) \Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} |a_k|\nonumber \\ & \geq &  \sum _{k=2}^\infty \eta \Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} |a_k|- \sum _{k=2}^\infty k \Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} |a_k|\nonumber \\ \end{eqnarray*}
</div>
<p> we deduce that </p>
<div class="displaymath" id="a0000000015">
  \begin{equation*}  \tfrac {1+\sum \limits _{k=2}^\infty k \big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k} |a_k|}{1+\sum \limits _{k=2}^\infty \big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k} |a_k|} {\gt} \eta . \end{equation*}
</div>
<p> Thus </p>
<div class="displaymath" id="a0000000016">
  \[  \Re \Bigg\{ \frac{z\big[\mathcal{K}^{m,r,s}_{\lambda _1,\lambda _2}f(z)\big]'}{\mathcal{K}^{m,r,s}_{\lambda _1,\lambda _2}f(z)}\Bigg\} {\gt}\eta , \qquad 0\leq \eta {\lt}1, z\in \mathcal{U}.  \]
</div>
<p>We note that the assertion (<a href="#co">13</a>) is sharp, moreover, the extremal function can be given by </p>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  f(z)=z+\sum _{k=2}^\infty \tfrac {(1-\eta )}{(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}z^k. \end{equation*}
</div>
<p><div class="corollary_thmwrapper " id="co1">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">4</span>
  </div>
  <div class="corollary_thmcontent">
  <p> If the hypotheses of Theorem <span class="rm"><a href="#coef">3</a></span> is satisfied, then </p>
<div class="equation" id="a0000000018">
<p>
  <div class="equation_content">
    \begin{equation}  |a_k| \leq \tfrac {(1-\eta )}{(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}},\qquad \forall k\geq 2. \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>

  </div>
</div> </p>
<h1 id="a0000000019">3 Closure Theorems</h1>
<p> Let the functions \(f_j(z)\) be defined by </p>
<div class="equation" id="clo">
<p>
  <div class="equation_content">
    \begin{equation} \label{clo} f_j(z)=z+\sum _{k=2}^\infty a_{k,j}z^{k};\quad \quad \quad (a_{k,j}\geq 0 , z \in \mathcal{U}). \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p> <div class="theorem_thmwrapper " id="closure">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5</span>
  </div>
  <div class="theorem_thmcontent">
  <p> Let the functions \(f_j(z)\) defined by <span class="rm">(<a href="#clo">13</a>)</span> be in the class \(\mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) for every \(j=1,2,...,l\). Then the function \(G(z)\) defined by </p>
<div class="displaymath" id="a0000000020">
  \begin{equation*}  G(z)= z+\sum _{k=2}^\infty b_{k}z^{k};\quad \quad \quad (b_{k}\geq 0 , z \in \mathcal{U}) \end{equation*}
</div>
<p> is a member of the class \(\mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\), where </p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  b_k=\tfrac {1}{l}\sum _{j=1}^l a_{k,j};\quad \quad \quad (k\geq 2). \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000022">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Since \(f_j(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\), from Theorem <a href="#coef">3</a> we can write </p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  \sum _{k=2}^\infty (k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} |a_{k,j}|\leq 1-\eta , \qquad 0\leq \eta \leq 1 \end{equation*}
</div>
<p> for every \(j=1,2,...,l\). Thus </p>
<div class="displaymath" id="a0000000024">
  \begin{equation*} \begin{split} & \sum _{k=2}^\infty (k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} |b_{k}|=\   \\ & = \sum _{k=2}^\infty (k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k}\bigg|\tfrac {1}{l}\sum _{j=1}^l a_{k,j}\bigg| \\ & \leq \tfrac {1}{l}\sum _{j=1}^l\Bigg(\sum _{k=n}{\infty }(k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} |a_{k,j}|\Bigg) \\ & = \tfrac {1}{l}\sum _{j=1}^l( 1-\eta )=( 1-\eta ). \end{split}\end{equation*}
</div>
<p> In view of Theorem <a href="#coef">3</a>, we conclude that \(G(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). </p>
<p><div class="theorem_thmwrapper " id="a0000000025">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>The class \( \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) is closed under convex linear combination. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(f_j(z)\) defined by (<a href="#clo">13</a>) be belonged to \(\mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) for every \(j=1,2,...,l\), it is sufficient to prove that the function </p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  h(z)=\mu f_1(z)+(1-\mu )f_2(z) \end{equation*}
</div>
<p> is also in the class \(\mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). </p>
<p>Let us write, for \(0 \leq \mu \leq 1\), </p>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  h(z)=z+\sum _{k=n}^\infty \{ \mu a_{k,1}+(1-\mu )a_{k,2}\} z^k, \end{equation*}
</div>
<p> we note that </p>
<div class="displaymath" id="a0000000029">
  \begin{align*} & \sum _{k=n}^{\infty }(k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k}\Big|\mu a_{k,1}+(1-\mu )a_{k,2} \Big|=\  \\ & \leq \sum _{k=n}^{\infty }(k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k}|\mu a_{k,1}|+\\ & \quad +\sum _{k=n}^{\infty }(k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k}|(1-\mu )a_{k,2}| \\ & =\mu \sum _{k=n}^{\infty }(k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k}|a_{k,1}|\\ & \quad +(1-\mu )\sum _{k=n}^{\infty }(k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k}|a_{k,2}|\\ & \leq \mu (1-\eta )+(1-\mu )(1-\eta )= (1-\eta ). \end{align*}
</div>
<p> It follows from Theorem <a href="#coef">3</a> that \( h(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\), which completes the proof. </p>
<p><div class="theorem_thmwrapper " id="a0000000030">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let </p>
<div class="displaymath" id="a0000000031">
  \begin{equation*}  f_0(z)=z \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  f_k(z)=z+\tfrac {(1-\eta )}{(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}z^k. \end{equation*}
</div>
<p> Then \( f(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) if and only if it can be expressed in the form </p>
<div class="equation" id="a0000000033">
<p>
  <div class="equation_content">
    \begin{equation}  f(z)=\sum _{k=0}^\infty \rho _kf_k(z) \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p> where \(\rho _k\geq 0\) and \(\sum _{k=0}^\infty \rho _k =1\). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000034">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Firstly, suppose that </p>
<div class="equation" id="a0000000035">
<p>
  <div class="equation_content">
    \begin{equation}  f(z)=\sum _{k=0}^\infty \rho _k f_k(z) \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p> where \(\rho _k\geq 0\) and \(\sum _{k=0}^\infty \rho _k =1\). Then </p>
<div class="displaymath" id="a0000000036">
  \begin{eqnarray*}  f(z) & =&  \sum _{k=0}^\infty \rho _k f_k(z) = \rho _0f_0(z)+\sum _{k=1}^\infty \rho _k f_k(z)\\ & =&  z+ \tfrac {(1-\eta )}{(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}\rho _{k} z^k. \end{eqnarray*}
</div>
<p> We observe that </p>
<div class="displaymath" id="a0000000037">
  \begin{equation*} \begin{split} & \sum _{k=n}^{\infty }(k-\eta )\Big[\tfrac {(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\Big] {\Gamma _k} \cdot \bigg[\tfrac {(1-\eta )}{(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}\bigg]\rho _{k}=\  \\ & = (1-\eta )\sum _{k=1}^\infty \rho _k = (1-\eta )(1-\rho _0)\leq (1-\eta ). \end{split}\end{equation*}
</div>
<p> In view of Theorem <a href="#coef">3</a>, we conclude that \(f(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). </p>
<p>Conversely, let us suppose that \(f(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). Since </p>
<div class="equation" id="a0000000038">
<p>
  <div class="equation_content">
    \begin{equation}  a_k \leq \tfrac {(1-\eta )}{(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}, \qquad \forall k\geq 2. \end{equation}
  </div>
  <span class="equation_label">22</span>
</p>
</div>
<p> Then by Corollary <a href="#co1">4</a>, we set </p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  \rho _k = \tfrac {(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta )}a_k, \end{equation*}
</div>
<p> and </p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  \rho _0 = 1- \sum _{k=1}^{\infty }\rho _k. \end{equation*}
</div>
<p> We thus conclude that \( f(z)=\sum _{k=0}^\infty \rho _k f_k(z)\). This completes the proof of the theorem. </p>
<h1 id="a0000000041">4 Convolution Properties</h1>
<p>For functions \(f_j(z)\in \mathcal{A};(j=1,2,...,m)\) given by </p>
<div class="displaymath" id="a0000000042">
  \begin{equation*}  f_j(z)= z+ \sum _{k=2}^{\infty }a_{k,j}z^k;\quad \quad \quad (z\in \mathcal{U}), \end{equation*}
</div>
<p> the Hadamard product (or convolution) of \(f_1(z),f_2(z),...,f_m(z)\) is defined by </p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  G_m(z)=(f_1\ast f_2\ast ...\ast f_m)(z)=z+\sum _{k=2}^\infty \Big( \textstyle \prod \limits _{j=1}^m a_{k,j}z^k\Big). \end{equation*}
</div>
<p> <div class="theorem_thmwrapper " id="a0000000044">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">8</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(f_j(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) for each \((j=1,2,...,m)\), then \(G_m(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) with </p>
<div class="displaymath" id="a0000000045">
  \begin{equation*}  \eta ^*=\tfrac {\prod \limits _{j=1}^m(1-\eta _j)}{\prod \limits _{j=1}^m(2-\eta _j)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}-\prod \limits _{j=1}^m(1-\eta _j)}. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000046">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>We use the mathematical induction to get to the required result. Firstly, we have to show that \(G_2(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\) for \(f_1(z)\) and \(f_2(z)\) belonging to \(\mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta _1)\) , \(\mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta _1)\) respectively. We can write </p>
<div class="displaymath" id="a0000000047">
  \begin{equation*}  \sum _{k=2}^\infty \tfrac {(k-\eta _j)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _j)} |a_{k,j}|\leq 1;\quad \quad \quad (j=1,2). \end{equation*}
</div>
<p> Applying the Schwarz inequality, we have the following inequality </p>
<div class="displaymath" id="a0000000048">
  \begin{equation*}  \sum _{k=2}^\infty \sqrt{\tfrac {(k-\eta _1)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _1)}\tfrac {(k-\eta _2)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _2)}} \sqrt{|a_{k,1}|\cdot |a_{k,2}|}\leq 1. \end{equation*}
</div>
<p> Then, we will determine the largest \(\eta ^*\) such that </p>
<div class="displaymath" id="a0000000049">
  \begin{equation*}  \sum _{k=2}^\infty \tfrac {(k-\eta ^*)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta ^*)} |a_{k,1}|\cdot |a_{k,2}|\leq 1. \end{equation*}
</div>
<p> That is </p>
<div class="displaymath" id="a0000000050">
  \begin{equation*} \begin{split} & \sum _{k=2}^\infty \tfrac {(k-\eta ^*)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta ^*)} |a_{k,1}|\cdot |a_{k,2}|\leq \  \\ & \leq \sum _{k=2}^\infty \sqrt{\tfrac {(k-\eta _1)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _1)}\tfrac {(k-\eta _2)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _2)}} \sqrt{|a_{k,1}||a_{k,2}|}. \end{split}\end{equation*}
</div>
<p> Therefore, we need to find the largest \(\eta ^*\) such that </p>
<div class="displaymath" id="a0000000051">
  \begin{equation*} \begin{split} & \tfrac {(k-\eta ^*)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta ^*)} \sqrt{|a_{k,1}|\cdot |a_{k,2}|}\leq \  \\ & \leq \sqrt{\tfrac {(k-\eta _1)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _1)}\tfrac {(k-\eta _2)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _2)}} \end{split}\end{equation*}
</div>
<p> for all \(k\geq 2\). Thus we can write </p>
<div class="displaymath" id="a0000000052">
  \begin{align*} & \tfrac {(k-\eta ^*)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta ^*)}\leq \\ & \leq \bigg\{ \tfrac {(k-\eta _1)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _1)}\bigg\} \bigg\{ \tfrac {(k-\eta _2)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta _2)}\bigg\} . \end{align*}
</div>
<p> After some calculations, we get </p>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  \eta ^* \leq 1- \tfrac {(k-1)(1-\eta _1)(1-\eta _2)} {(k-\eta _1)(k-\eta _2)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big]\Gamma _k-(1-\eta _1)(1-\eta _2)}. \end{equation*}
</div>
<p> We note that the right hand side of the above inequality is an increasing function for all \(k\geq 2\). This implies that </p>
<div class="displaymath" id="*">
  \begin{align} \label{*} \eta ^* =&  \min _{k\geq 2}\bigg\{ \tfrac {(k-1)(1-\eta _1)(1-\eta _2)} {(k-\eta _1)(k-\eta _2)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big]\Gamma _k-(1-\eta _1)(1-\eta _2)}\bigg\}  \nonumber \\ =&  \tfrac {(1-\eta _1)(1-\eta _2)} {(2-\eta _1)(2-\eta _2)\big[\frac{(1+\lambda _1)^{m-1}}{(1+\lambda _2)^m}\big]\Gamma _2-(1-\eta _1)(1-\eta _2)}. \end{align}
</div>
<p> Thus \(G_2(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). Therefore the theorem is true for \(m=2\). Now, we suppose that \(G_{m-1}(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta _0)\) and \(f_m(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta _m)\), where </p>
<div class="displaymath" id="a0000000054">
  \begin{equation*}  \eta _0 = \tfrac {\prod \limits _{j=1}^{m-1}(1-\eta _j)}{\prod \limits _{j=1}^{m-1}(2-\eta _j)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^{m-1}}\big] {\Gamma _k}-\prod \limits _{j=1}^{m-1}(1-\eta _j)}. \end{equation*}
</div>
<p> Replacing \(\eta _1\) by \(\eta _0\), and \(\eta _2\) by \(\eta _m\) in the inequality (<a href="#*">27</a>), we get </p>
<div class="displaymath" id="a0000000055">
  \begin{align*} \label{*} \eta ^* =&  \tfrac {(1-\eta _0)(1-\eta _m)} {(2-\eta _0)(2-\eta _m)\big[\frac{(1+\lambda _1)^{m-1}}{(1+\lambda _2)^m}\big]\Gamma _2-(1-\eta _0)(1-\eta _m)}\  \\ =&  \tfrac {\prod \limits _{j=1}^m(1-\eta _j)}{\prod \limits _{j=1}^m(2-\eta _j)\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}-\prod \limits _{j=1}^m(1-\eta _j)}. \end{align*}
</div>
<p> For the integer \(m\) the theorem ia also true. By the mathematical induction, the proof of the theorem is complete. </p>
<h1 id="a0000000056">5 Integral Operator</h1>
<p> In this section we consider integral transforms of functions in the class \( \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). <div class="theorem_thmwrapper " id="a0000000057">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">9</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let the function \(f(z)\) defined by <span class="rm">(<a href="#one">1</a>)</span> be in the class \( \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\). Then the integral transforms </p>
<div class="equation" id="a0000000058">
<p>
  <div class="equation_content">
    \begin{equation}  G_c(z)=c\int _0^1u^{c-2}f(uz)du; \quad \quad (c>0) \end{equation}
  </div>
  <span class="equation_label">28</span>
</p>
</div>
<p> are in the class \( \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\gamma )\), where </p>
<div class="equation" id="a0000000059">
<p>
  <div class="equation_content">
    \begin{equation}  \gamma =1-\tfrac {c(1-\eta )}{(2-\eta )(c+1)-c(1-\eta )}. \end{equation}
  </div>
  <span class="equation_label">29</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000060">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(f(z)=z+\sum _{k=2}^{\infty } a_k z^k\). Then we have </p>
<div class="equation" id="a0000000061">
<p>
  <div class="equation_content">
    \begin{equation}  G_c(z)=c\int _0^1u^{c-2}f(uz)du = z+\sum _{k=2}^{\infty }\Big( \tfrac {c}{c+k-1}\Big) a_k z^k. \end{equation}
  </div>
  <span class="equation_label">30</span>
</p>
</div>
<p> Since \(f(z)\in \mathcal{S}^{m,r,s}_{\lambda _1,\lambda _2}(\eta )\), we have </p>
<div class="equation" id="a0000000062">
<p>
  <div class="equation_content">
    \begin{equation}  \sum _{k=2}^\infty \tfrac {(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta )} |a_{k}|\leq 1. \end{equation}
  </div>
  <span class="equation_label">31</span>
</p>
</div>
<p> In view of Theorem <a href="#coef">3</a>, we shall find the largest \(\gamma \) for which </p>
<div class="equation" id="a0000000063">
<p>
  <div class="equation_content">
    \begin{equation}  \sum _{k=2}^\infty \tfrac {(k-\gamma )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\gamma ) \big( \frac{c}{c+k-1}\big)}|a_{k}|\leq 1. \end{equation}
  </div>
  <span class="equation_label">32</span>
</p>
</div>
<p> Let us find the range of values of \(\gamma \) for which </p>
<div class="displaymath" id="a0000000064">
  \begin{equation*}  \tfrac {(k-\gamma )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\gamma )\big( \frac{c}{c+k-1}\big)} \leq \tfrac {(k-\eta )\big[\frac{(1+\lambda _1(k-1))^{m-1}}{(1+\lambda _2(k-1))^m}\big] {\Gamma _k}}{(1-\eta )}, \qquad (k\geq 2). \end{equation*}
</div>
<p>After some calculations, we obtain from the above inequality that </p>
<div class="displaymath" id="a0000000065">
  \begin{equation*}  \gamma \leq 1-\tfrac {c(k-1)(1-\eta )}{(k-\eta )(c+k-1)-c(1-\eta )}. \end{equation*}
</div>
<p> We note that the right hand side of the above inequality is an increasing function for all \(k\geq 2\). This implies that </p>
<div class="displaymath" id="a0000000066">
  \begin{eqnarray*}  \gamma & =&  \min _{k\geq 2}\bigg\{ 1-\tfrac {c(k-1)(1-\eta )}{(k-\eta )(c+k-1)-c(1-\eta )}\bigg\} \nonumber \\ & =&  1-\tfrac {c(1-\eta )}{(2-\eta )(c+1)-c(1-\eta )}. \end{eqnarray*}
</div>
<p> The proof is complete. </p>
<p><div class="acknowledgement_thmwrapper " id="a0000000067">
  <div class="acknowledgement_thmheading">
    <span class="acknowledgement_thmcaption">
    Acknowledgements
    </span>
  </div>
  <div class="acknowledgement_thmcontent">
  <p>The authors wish to thank the referee for the comments to improve the contents of the manuscript. </p>

  </div>
</div> </p>

<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="dziok">1</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1016/S0096-3003(98)10042-5"> <i class="sc">J. Dziok</i> and <i class="sc">H. M. Srivastava</i>, <i class="it">Classes of analytic functions associated with the generalized hypergeometric function</i>, Appl. Math. Comput., <b class="bf">103</b>(1) (1999), pp.&#160;1–13. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="abbadi">2</a></dt>
  <dd><p><i class="sc">M. H. Al-Abbadi</i> and <i class="sc">M. Darus</i>, <i class="it">Differential subordination for new generalized derivative operator</i>, Acta Univ. Apulensis, <b class="bfseries">20</b> (2009), pp.&#160; 265–280. </p>
</dd>
  <dt><a name="hohlov">3</a></dt>
  <dd><p><i class="sc">Yu. E. Hohlov</i>, <i class="it">Operators and operations in the class of univalent functions</i>, Izv. Vyssh. Uchebn. Zaved. Mat., <b class="bfseries">10</b>(10) (1978), pp.&#160; 83–89. </p>
</dd>
  <dt><a name="carlson">4</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1137/0515057"> <i class="sc">B. C. Carlson</i> and <i class="sc">D. B. Shaffer</i>, <i class="it">Starlike and prestarlike hypergeometric function</i>, SIAM J. Math. Anal., <b class="bfseries">15</b>(4) (1984), pp.&#160;737–745, 1984. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="ruscheweyh">5</a></dt>
  <dd><p><a href ="http://dx.doi.org/10.1090/S0002-9939-1975-0367176-1"> <i class="sc">S. Ruscheweyh</i>, <i class="it">New criteria for univalent functions</i>, Proc. Amer. Math. Soc., <b class="bfseries">49</b> (1975), pp.&#160; 109–115. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="shaqsi">6</a></dt>
  <dd><p><i class="sc">K. Al-Shaqsi</i> and <i class="sc">M. Darus</i>, <i class="it">On univalent functions with respect to the k-symmetric points defined by a generalization Ruscheweyh derivative operators</i>, Jour. Anal. Appl., <b class="bfseries">7</b>(1) (2009), pp.&#160; 53–61. </p>
</dd>
  <dt><a name="darus">7</a></dt>
  <dd><p><i class="sc">K. Al-Shaqsi</i> and <i class="sc">M. Darus</i>, <i class="it">Differential Subordination with generalized derivative operator</i>, Int. J. Comp. Math. Sci., <b class="bfseries">2</b>(2) (2008), pp.&#160; 75–78. </p>
</dd>
  <dt><a name="owa">8</a></dt>
  <dd><p><i class="sc">J. Nishiwaki</i> and <i class="sc">S. Owa</i>, <i class="it">Convolutions for certain analytic functions</i>, Gen. Math., <b class="bfseries">15</b> (2007), no. 2–3, pp.&#160; 38–51. </p>
</dd>
</dl>


</div>
</div> <!--main-text -->
</div> <!-- content-wrapper -->
</div> <!-- content -->
</div> <!-- wrapper -->

<nav class="prev_up_next">
</nav>

<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/jquery.min.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/plastex.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/svgxuse.js"></script>
</body>
</html>