%% This document created by Scientific Word (R) Version 2.5
%% Starting shell: article


\documentclass[12pt,thmsb]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{sw20lart}

\setcounter{MaxMatrixCols}{10}
%TCIDATA{TCIstyle=article/art4.lat,lart,article}

%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.50.0.2953}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%TCIDATA{Created=Mon Jan 28 16:18:26 2008}
%TCIDATA{LastRevised=Thursday, August 16, 2012 16:06:44}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{Language=American English}

\input{tcilatex}
\begin{document}

\author{A. O. Mostafa$^{1}$, M. K. Aouf$^{2}$, A. Shamandy$^{3}$ and E. A.
Adwan$^{4}$ \\
%EndAName
\\
Department of Mathematics, Faculty of Science, \\
Mansoura University, Mansoura 35516, Egypt\\
$^{1}$adelaeg254@yahoo.com,\ \ $\ \ \ ^{2}$mkaouf127@yahoo.com\\
$^{3}$\ shamandy16@hotmail.com,\ \ $^{4}$eman.a2009@yahoo.com}
\title{ \textbf{Some Subordination and Superordination for the Wright
Generalized Hypergeometric Function}}
\date{}
\maketitle

\begin{abstract}
In this paper, we obtain some subordination and superordination results for
the Wright generalized hypergeometric function. Sandwich-type theorem for
these multivalent function is also obtained.
\end{abstract}

\noindent \textbf{Keywords and phrases}: p-Valent functions, subordination,
superordination, Wright generalized hypergeometric function.

\noindent 2000 Mathematics Subject Classification: 30C45.\newline

\noindent \bigskip {\LARGE 1.\quad Introduction}

Let $H(U)$ be the class of functions analytic in $U=\{z\in 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
:|z|<1\}$ and $H[a,n]$ be the subclass of $H(U)$ consisting of functions of
the form $f(z)=a+a_{n}z^{n}+$ $a_{n+1}z^{n+1}+...$, with $H_{0}=H[0,1]$ and $%
H=H[1,1]$. Let $A\left( p\right) $ denote the class of all analytic
functions of the form

\begin{equation}
f(z)=z^{p}+\sum_{n=1+p}^{\infty }a_{n}z^{n}\ \ \ \left( p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
=\left\{ 1,2,3,...\right\} ;z\in U\right) .  \tag{1.1}
\end{equation}%
Let $f$ and $F$ be members of $H(U)$. The function $f(z)$ is said to be
subordinate to $F(z)$, or $F(z)$ is superordinate to $f(z)$, if there exists
a function $\omega (z)$ analytic in $U$ with $\omega (0)=0$ and $|\omega
(z)|<1(z\in U)$, such that $f(z)=F(\omega (z))$. In such a case we write $%
f(z)\prec F(z)$. If $F$ is univalent, then $f(z)\prec F(z)$ if and only if $%
f(0)=F(0)$ and $f(U)\subset F(U)$ $\left( \text{see }\left[ 12\right] \text{
and }\left[ 13\right] \right) $.

Let $\phi :%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{2}\times U\rightarrow 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$ and $h\left( z\right) $ be univalent in $U.$ If $p\left( z\right) $ is
analytic in $U$ and satisfies the first order differential subordination:%
\begin{equation}
\phi \left( p\left( z\right) ,zp^{^{\prime }}\left( z\right) ;z\right) \prec
h\left( z\right) ,  \tag{1.2}
\end{equation}%
then $p\left( z\right) $ is a solution of the differential subordination $%
(1.2)$. The univalent function $q\left( z\right) $ is called a dominant of
the solutions of the differential subordination $(1.2)$ if $p\left( z\right)
\prec q\left( z\right) $ for all $p\left( z\right) $ satisfying $(1.2)$. A
univalent dominant $\tilde{q}$ that satisfies $\tilde{q}\prec q$ for all
dominants of $(1.2)$ is called the best dominant. If $p\left( z\right) $ and 
$\phi \left( p\left( z\right) ,zp^{^{\prime }}\left( z\right) ;z\right) $
are univalent in $U$ and if $p(z)$ satisfies first order differential
superordination:%
\begin{equation}
h\left( z\right) \prec \phi \left( p\left( z\right) ,zp^{^{\prime }}\left(
z\right) ;z\right) ,  \tag{1.3}
\end{equation}%
then $p\left( z\right) $ is a solution of the differential superordination $%
(1.3)$. An analytic function $q\left( z\right) $ is called a subordinant of
the solutions of the differential superordination $(1.3)$ if $q\left(
z\right) \prec p\left( z\right) $ for all $p\left( z\right) $ satisfying $%
(1.3)$. A univalent subordinant $\tilde{q}$ that satisfies $q\prec \tilde{q}$
for all subordinants of $(1.3)$ is called the best subordinant $\left( \text{%
see }\left[ 12\right] \text{ and }\left[ 13\right] \right) $.\newline

\noindent For analytic functions $f\left( z\right) \in A(p),$ given by (1.1)
and $\phi \left( z\right) \in A(p)$ given by $\phi \left( z\right)
=z^{p}+\sum\limits_{n=1+p}^{\infty }b_{n}z^{n}\ \ \left( p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) $, the Hadamard product (or convolution) of $f\left( z\right) $
\noindent and $\phi \left( z\right) $, is defined by

\begin{equation}
\left( f\ast \phi \right) \left( z\right) =z^{p}+\sum_{n=1+p}^{\infty
}a_{n}b_{n}z^{n}=\left( \phi \ast f\right) \left( z\right) .  \tag{1.5}
\end{equation}

Let $\alpha _{1},A_{1},...,\alpha _{q},A_{q}$\ and $\beta
_{1},B_{1},...,\beta _{s},B_{s}$ $\left( q,s\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) $ be positive real parameters such that

\begin{equation*}
1+\sum_{j=1}^{s}B_{j}-\sum_{j=1}^{q}A_{j}\geq 0.
\end{equation*}

\noindent The Wright generalized hypergeometric function [21] (see also [20])

\begin{equation*}
_{q}\Psi _{s}\left[ \left( \alpha _{1},A_{1}\right) ,...,\left( \alpha
_{q},A_{q}\right) ;\left( \beta _{1},B_{1}\right) ,...,\left( \beta
_{s},B_{s}\right) ;z\right] =_{q}\Psi _{s}\left[ \left( \alpha
_{i},A_{i}\right) _{1,q};\left( \beta _{i},B_{i}\right) _{1,s};z\right]
\end{equation*}%
\noindent is defined by%
\begin{equation*}
_{q}\Psi _{s}\left[ \left( \alpha _{i},A_{i}\right) _{1,q};\left( \beta
_{i},B_{i}\right) _{1,s};z\right] =\dsum\limits_{n=0}^{\infty }\frac{%
\tprod\limits_{i=1}^{q}\Gamma \left( \alpha _{i}+nA_{i}\right) }{%
\prod\limits_{i=1}^{s}\Gamma \left( \beta _{i}+nB_{i}\right) }.\frac{z^{n}}{%
n_{!}}\ \ \ \left( z\in U\right) .
\end{equation*}%
\noindent If $A_{i}=1(i=1,...,q)$ and $B_{i}=1\left( i=1,...,s\right) ,$ we
have the relationship:

\begin{equation*}
\Omega _{q}\Psi _{s}\left[ \left( \alpha _{i},1\right) _{1,q};\left( \beta
_{i},1\right) _{1,s};z\right] =\text{ }_{q}F_{s}\left( \alpha
_{1},...,\alpha _{q};\beta _{_{1}},...,\beta _{s};z\right) ,
\end{equation*}%
\noindent where $_{q}F_{s}\left( \alpha _{1},...,\alpha _{q};\beta
_{_{1}},...,\beta _{s};z\right) $ is the generalized hypergeometric function
(see [20]) and

\begin{equation}
\Omega =\frac{\prod\limits_{i=1}^{s}\Gamma \left( \beta _{i}\right) }{%
\prod\limits_{i=1}^{q}\Gamma \left( \alpha _{i}\right) }.  \tag{1.6}
\end{equation}

\noindent The Wright generalized hypergeometric functions were invoked \ in
the geometric function theory (see [16] and [17]).

\noindent By using the generalized hypergeometric function Dziok and
Srivastava [7] introduced a linear operator. In [6] Dziok and Raina and in
[2] Aouf\ and Dziok \noindent extended this linear operator by using Wright
generalized hypergeometric function.

\noindent Aouf et al. [3] considered the following linear operator

\begin{equation*}
\theta _{p,q,s}\left[ \left( \alpha _{i},A_{i}\right) _{1,q};\left( \beta
_{i},B_{i}\right) _{1,s}\right] :A(p)\rightarrow A(p),
\end{equation*}

\noindent defined by the following Hadamard product:%
\begin{equation*}
\theta _{p,q,s}\left[ \left( \alpha _{i},A_{i}\right) _{1,q};\left( \beta
_{i},B_{i}\right) _{1,s}\right] f\left( z\right) =_{q}\Phi _{s}^{p}\left[
\left( \alpha _{i},A_{i}\right) _{1,q};\left( \beta _{i},B_{i}\right)
_{1,s};z\right] \ast f\left( z\right) ,
\end{equation*}

\noindent where $_{q}\Phi _{s}^{p}\left[ \left( \alpha _{i},A_{i}\right)
_{1,q};\left( \beta _{i},B_{i}\right) _{1,s};z\right] $ is given by

\begin{equation*}
_{q}\Phi _{s}^{p}\left[ \left( \alpha _{i},A_{i}\right) _{1,q};\left( \beta
_{i},B_{i}\right) _{1,s};z\right] =\Omega \text{ }z_{\text{ \ \ }q}^{p}\Psi
_{s}\left[ \left( \alpha _{i},A_{i}\right) _{1,q};\left( \beta
_{i},B_{i}\right) _{1,s};z\right] .
\end{equation*}%
\noindent We observe that, for a function $f\left( z\right) $ of the form $%
\left( 1.1\right) ,$ we have%
\begin{equation}
\theta _{p,q,s}\left[ \left( \alpha _{i},A_{i}\right) _{1,q};\left( \beta
_{i},B_{i}\right) _{1,s}\right] f\left( z\right) =z^{p}+\sum_{n=1+p}^{\infty
}\Omega \sigma _{n,p}\left( \alpha _{1}\right) a_{n}z^{n},  \tag{1.7}
\end{equation}%
\noindent where $\Omega $ is given by $\left( 1.6\right) $ and $\sigma
_{n,p}\left( \alpha _{1}\right) $ is defined by

\begin{equation}
\sigma _{n,p}\left( \alpha _{1}\right) =\frac{\Gamma \left( \alpha
_{1}+A_{1}\left( n-p\right) \right) ...\Gamma \left( \alpha _{q}+A_{q}\left(
n-p\right) \right) }{\Gamma \left( \beta _{1}+B_{1}\left( n-p\right) \right)
...\Gamma \left( \beta _{s}+B_{s}\left( n-p\right) \right) \left( n-p\right)
!}.  \tag{1.8}
\end{equation}%
\noindent If, for convenience, we write%
\begin{equation*}
\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right)
=\theta _{p,q,s}\left[ \left( \alpha _{1},A_{1}\right) ,...,\left( \alpha
_{q},A_{q}\right) ;\left( \beta _{1},B_{1}\right) ,...,\left( \beta
_{s},B_{s}\right) \right] f\left( z\right) ,
\end{equation*}%
\noindent then one can easily verify from $\left( 1.7\right) $ that

\begin{equation*}
zA_{1}\left( \theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left(
z\right) \right) ^{\prime }=\alpha _{1}\theta _{p,q,s}\left[ \alpha
_{1}+1,A_{1},B_{1}\right] f\left( z\right)
\end{equation*}

\begin{equation}
-(\alpha _{1}-pA_{1})\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right]
f\left( z\right) \ (A_{1}>0).  \tag{1.9}
\end{equation}%
\newline

For $p=1$, $\theta _{1,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] =\theta %
\left[ \alpha _{1}\right] $ which\ was introduced by Dziok and Raina [6] and
studied by \noindent Aouf and Dziok [2]. We note that, for $f\left( z\right)
\in A(p),$ $A_{i}=1\left( i=1,2,...,q\right) $ and $B_{i}=1\left(
i=1,2,...,s\right) ,$ we obtain $\theta _{p,q,s}\left[ \alpha _{1},1,1\right]
f\left( z\right) =H_{p,q,s}[\alpha _{1}]f\left( z\right) ,$ where $%
H_{p,q,s}[\alpha _{1}]$ is \noindent the Dziok-Srivastava operator (see [7]).

We note also that, for $f\left( z\right) \in A(p),q=2$, $s=1$ and$\
A_{1}=A_{2}=B_{1}=1$, we have:

\noindent (1) $\theta _{p,2,1}\left[ a,1;c\right] f\left( z\right)
=L_{p}\left( a,c\right) f\left( z\right) $ $\ \left( a>0,c>0,p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) $ \ (see [18]);\newline

\noindent (2) $\theta _{p,2,1}\left[ \mu +p,1;1\right] f\left( z\right)
=D^{\mu +p-1}f\left( z\right) $ \ $\left( \mu >-p,p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) ,$ where $D^{\mu +p-1}f\left( z\right) $ is the $\left( \mu
+p-1\right) -$the order Ruscheweyh derivative (see [8]);\newline

\noindent (3) $\theta _{p,2,1}\left[ \nu +p,1;\nu +p+1\right] f\left(
z\right) =F_{\nu ,p}(f)(z)$ \ \ $\left( \nu >-p,p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) $, where $F_{\nu ,p}\left( f\right) \left( z\right) $ is the
generalized Bernardi-Libera-Livingston-integral operator (see \noindent
\lbrack 5]);\newline

\noindent (4) $\theta _{p,2,1}\left[ c,1;a\right] f\left( z\right)
=I_{c,p}^{a}f\left( z\right) $ $\left( a\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,c\in 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
\backslash 
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{0}^{-},p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) ,$where the operator $I_{c,p}^{a}$ was introduced and studied by
AL-Kharasani and Al-Hajiry (see \noindent \lbrack 1])$;$\newline

\noindent (5) $\theta _{p,2,1}\left[ p+1,1;n+p\right] f\left( z\right)
=I_{n,p}f\left( z\right) $ $\left( n\in 
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
;n>-p,p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) ,$where the operator $I_{n,p}$ was introduced and studied by Liu and
Noor (see [9]);\newline

\noindent (6) $\theta _{p,2,1}\left[ \lambda +p,c;a\right] f\left( z\right)
=I_{p}^{\lambda }\left( a,c\right) f\left( z\right) $ \ $\left( a,c\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\backslash 
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
_{o}^{-};\lambda >-p,p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) ,$ where $I_{p}^{\lambda }\left( a,c\right) $ is the
Cho-Kwon-Srivastava operator (see [4]);\newline

\noindent (7) $\theta _{p,2,1}\left[ 1+p,1;1+p-\mu \right] f\left( z\right)
=\Omega _{z}^{\left( \mu ,p\right) }f\left( z\right) $ $\ \left( -\infty
<\mu <1+p,p\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) ,$ where the operator $\Omega _{z}^{\left( \mu ,p\right) }$ was
introduced and studied by Patel and Mishra (see \noindent \lbrack 14]) and
studied by Srivastava and Aouf [19] when $\left( 0\leq \mu <1\right) $.%
\newline

To prove our results, we need the following definitions and lemmas.

\noindent \textbf{Definition 1} $\left[ 12\right] .$ \textit{Denote by }$%
\tciFourier $\textit{\ the set of all functions }$q(z)$\textit{\ that are
analytic and injective on }$\bar{U}\backslash E(q)$\textit{\ where}%
\begin{equation*}
E(q)=\left\{ \zeta \in \partial U:\lim_{z\rightarrow \zeta }q(z)=\infty
\right\} ,
\end{equation*}%
\textit{\noindent and are such that }$q^{^{\prime }}(\zeta )\neq 0$\textit{\
for }$\zeta \in \partial U\backslash E(q)$\textit{. Further let the subclass
of }$\tciFourier $\textit{\ for which }$q(0)=a$\textit{\ be denoted by }$%
\tciFourier (a)$\textit{, }$\tciFourier (0)\equiv \tciFourier _{0}$\textit{\
and }$\tciFourier (1)\equiv \tciFourier $\textit{.}

\noindent \textbf{Definition 2} $\left[ 13\right] $. \textit{A function }$%
L\left( z,t\right) $\textit{\ }$\left( z\in U,t\geq 0\right) $\textit{\ is
said to be a subordination chain if }$L\left( 0,t\right) $\textit{\ is
analytic and univalent in }$U$\textit{\ for all }$t\geq 0,L\left( z,0\right) 
$\textit{\ is \noindent continuously differentiable on }$\left[ 0;1\right) $%
\textit{\ for all }$z\in U$\textit{\ and }$L\left( z,t_{1}\right) \prec
L\left( z,t_{2}\right) $\textit{\ for all }$0\leq t_{1}\leq t_{2}$\textit{.}

\noindent \textbf{Lemma 1 }$\left[ 15\right] $\textbf{.} \textit{The
function }$L\left( z,t\right) :U\times $\textit{\ }$\left[ 0;1\right)
\longrightarrow 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$\textit{\ of the form}%
\begin{equation*}
L\left( z,t\right) =a_{1}\left( t\right) z+a_{2}\left( t\right) z^{2}+...\ \
\ \left( a_{1}\left( t\right) \neq 0;t\geq 0\right) ,
\end{equation*}%
\textit{\noindent and }$\lim_{t\rightarrow \infty }\left\vert a_{1}\left(
t\right) \right\vert =\infty $\textit{\ is a subordination chain if and only
if}%
\begin{equation*}
\func{Re}\left\{ \frac{z\partial L\left( z,t\right) /\partial z}{\partial
L\left( z,t\right) /\partial t}\right\} >0\ \ \ \left( z\in U,t\geq 0\right)
.
\end{equation*}

\noindent \textbf{Lemma 2 }$\left[ 10\right] $. \textit{Suppose that the
function }$H:%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{2}\rightarrow 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$\textit{\ satisfies the condition}

\begin{equation*}
\func{Re}\left\{ H\left( is;t\right) \right\} \leq 0
\end{equation*}%
\textit{\noindent for all real }$s$\textit{\ and for all }$t\leq -n\left(
1+s^{2}\right) /2$\textit{, }$n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
.$\textit{If the function }$p(z)=1+p_{n}z^{n}+p_{n+1}z^{n+1}+...$\textit{\
is analytic in }$U$\textit{\ and }%
\begin{equation*}
\func{Re}\left\{ H\left( p(z);zp^{^{\prime }}(z)\right) \right\} >0\ \ \ \
\left( z\in U\right) ,
\end{equation*}%
\textit{\noindent then }$\func{Re}\left\{ p(z)\right\} >0$\textit{\ for }$%
z\in U.$

\noindent \textbf{Lemma 3} $\left[ 11\right] $. \textit{Let }$\kappa ,\gamma
\in 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$\textit{\ with }$\kappa \neq 0$\textit{\ and let }$h\in H(U)$\textit{\ with 
}$h(0)=c$\textit{. If }$\func{Re}\left\{ \kappa h(z)+\gamma \right\}
>0\left( z\in U\right) ,$\textit{\ then the solution of the following
differential \noindent equation:}%
\begin{equation*}
q\left( z\right) +\frac{zq^{^{\prime }}\left( z\right) }{\kappa q(z)+\gamma }%
=h\left( z\right) ~\ \left( z\in U;q(0)=c\right)
\end{equation*}%
\textit{\noindent is analytic in }$U$\textit{\ and satisfies }$\func{Re}%
\left\{ \kappa q(z)+\gamma \right\} >0$\textit{\ for }$z\in U$\textit{.}

\noindent \textbf{Lemma 4} $\left[ 10\right] $. \textit{Let }$p\in
\tciFourier (a)$\textit{\ and let }$q(z)=a+a_{n}z^{n}+a_{n+1}z^{n+1}+...$%
\textit{be analytic in }$U$\textit{\ with }$q\left( z\right) \neq a$\textit{%
\ and }$n\geq 1$\textit{. If }$q$\textit{\ is not subordinate to }$p$\textit{%
, then there \noindent exists two points }$z_{0}=r_{0}e^{i\theta }\in U$%
\textit{\ and }$\zeta _{0}\in \partial U\backslash E(q)$\textit{\ such that}%
\begin{equation*}
q(U_{r_{0}})\subset p(U);\ \ \ q(z_{_{0}})=p(\zeta _{0})\ \ \ \text{ and \ \ 
}z_{0}p^{^{\prime }}(z_{0})=m\zeta _{0}p^{^{\prime }}(\zeta _{0})\ \ \
\left( m\geq n\right) .
\end{equation*}

\noindent \textbf{Lemma 5} $\left[ 13\right] $. \textit{Let }$q\in H[a,1]$%
\textit{\ and }$\phi :%
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
^{2}\rightarrow 
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$\textit{. Also set }$\phi \left( q\left( z\right) ,zq^{^{\prime }}\left(
z\right) \right) =h\left( z\right) .$\textit{\ If }$L\left( z,t\right) =\phi
\left( q\left( z\right) ,tzq^{^{\prime }}\left( z\right) \right) $\textit{\
is a subordination chain and \noindent }$q\in H[a,1]\cap \tciFourier (a)$%
\textit{, then}%
\begin{equation*}
h\left( z\right) \prec \varphi \left( p\left( z\right) ,zp^{^{\prime
}}\left( z\right) \right)
\end{equation*}%
\textit{\noindent implies that }$q\left( z\right) \prec p\left( z\right) $%
\textit{. Furthermore, if }$\varphi \left( q\left( z\right) ,zq^{^{\prime
}}\left( z\right) \right) =h\left( z\right) $\textit{\ has a univalent
solution }$q\in \tciFourier (a)$\textit{, then }$q$\textit{\ is the best
subordinant.\newline
}

\noindent {\LARGE 2.\quad Main results}\newline

Unless otherwise mentioned, we shall assume in the reminder of this paper
that, the parameters, $\eta >0,$ $\alpha _{1},A_{1},...,\alpha _{q},A_{q}$\
and $\beta _{1},B_{1},...,\beta _{s},B_{s}$ $\left( q,s\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
\right) $ are \noindent positive real numbers and $z\in U$.

\noindent \textbf{Theorem 1. }\textit{Let }$f,g\in A(p)$\textit{and }%
\begin{equation}
\func{Re}\left\{ 1+\tfrac{z\phi ^{^{\prime \prime }}\left( z\right) }{\phi
^{^{\prime }}\left( z\right) }\right\} >-\delta \ \ \ \ \ \ \left( \phi
\left( z\right) =\left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1}+1,A_{1},B_{1}\right] g\left( z\right) }{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] g\left( z\right) }\right) \left( \tfrac{\theta
_{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] g\left( z\right) }{z^{p}}%
\right) ^{\eta }\right) ,  \tag{2.1}
\end{equation}%
\textit{where }$\delta $\textit{\ is given by}%
\begin{equation}
\delta =\frac{A_{1}^{2}+\left( \eta \alpha _{1}\right) ^{2}-\left\vert
A_{1}^{2}-\left( \eta \alpha _{1}\right) ^{2}\right\vert }{4\eta \alpha
_{1}A_{1}}\qquad \left( z\in U\right) .  \tag{2.2}
\end{equation}%
\textit{Then the subordination condition:}%
\begin{equation}
\left( \tfrac{\theta _{p,q,s}\left[ \alpha _{1}+1,A_{1},B_{1}\right] f\left(
z\right) }{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left(
z\right) }\right) \left( \tfrac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}%
\right] f\left( z\right) }{z^{p}}\right) ^{\eta }\prec \phi \left( z\right) ,
\tag{2.3}
\end{equation}%
\textit{implies that}%
\begin{equation}
\left( \frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left(
z\right) }{z^{p}}\right) ^{\eta }\prec \left( \frac{\theta _{p,q,s}\left[
\alpha _{1},A_{1},B_{1}\right] g\left( z\right) }{z^{p}}\right) ^{\eta }, 
\tag{2.4}
\end{equation}%
\textit{where }$\left( \frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}%
\right] g\left( z\right) }{z^{p}}\right) ^{\eta }$\textit{\ is the best
dominant.}

\noindent \textbf{Proof.} Let us define the functions $F(z)$ and $G(z)$ in $%
U $ by%
\begin{equation}
F(z)=\left( \frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right]
f\left( z\right) }{z^{p}}\right) ^{\eta }\text{ \ \ \ and \ \ \ }G(z)=\left( 
\frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] g\left( z\right) 
}{z^{p}}\right) ^{\eta },\ \   \tag{2.5}
\end{equation}%
we assume here, without loss of generality, that $G(z)$\ is analytic,
univalent on $\bar{U}$ and 
\begin{equation*}
G^{^{\prime }}(\zeta )\neq 0\ \ \ \ \ \left( \left\vert \zeta \right\vert
=1\right) .
\end{equation*}%
If not, then we replace $F(z)$ and $G(z)$ by $F(\rho z)$ and $G(\rho z)$,
respectively, with $0<\rho <1.$ These new functions have the desired
properties on $\bar{U}$, so we can use them in the proof of our result and
the results would follow by letting $\rho \rightarrow 1.$

We first show that, if%
\begin{equation}
q\left( z\right) =1+\frac{zG^{^{\prime \prime }}\left( z\right) }{%
G^{^{\prime }}\left( z\right) }\ \ \ \ ,  \tag{2.6}
\end{equation}%
then%
\begin{equation*}
\func{Re}\left\{ q\left( z\right) \right\} >0.
\end{equation*}%
From $\left( 1.9\right) $ and the definition of the functions $G,\phi ,$ we
obtain that%
\begin{equation}
\phi \left( z\right) =G\left( z\right) +\frac{A_{1}}{\eta \alpha _{1}}%
zG^{^{\prime }}\left( z\right) .  \tag{2.7}
\end{equation}%
Differentiating both sides of $\left( 2.7\right) $ with respect to $z$ yields%
\begin{equation}
\phi ^{^{\prime }}\left( z\right) =\frac{A_{1}+\eta \alpha _{1}}{\eta \alpha
_{1}}G^{^{\prime }}\left( z\right) +\frac{A_{1}}{\eta \alpha _{1}}%
zG^{^{^{\prime \prime }}}\left( z\right) \ .  \tag{2.8}
\end{equation}%
Combining $\left( 2.6\right) $ and $\left( 2.8\right) $, we easily get%
\begin{equation}
1+\frac{z\phi ^{^{\prime \prime }}\left( z\right) }{\phi ^{^{\prime }}\left(
z\right) }=q\left( z\right) +\frac{zq^{^{\prime }}\left( z\right) }{q\left(
z\right) +\frac{\eta \alpha _{1}}{A_{1}}}=h(z)\ \ \ \ \left( z\in U\right) .
\tag{2.9}
\end{equation}%
It follows from $\left( 2.1\right) $ and $\left( 2.9\right) $ that%
\begin{equation}
\func{Re}\left\{ h\left( z\right) +\frac{\eta \alpha _{1}}{A_{1}}\right\}
>0\ \ \ \left( z\in U\right) .  \tag{2.10}
\end{equation}%
Moreover, by using Lemma 3, we conclude that the differential equation $%
\left( 2.9\right) $\ has a solution $q\left( z\right) \in H\left( U\right) $
with $h\left( 0\right) =q\left( 0\right) =1$. Let%
\begin{equation*}
H\left( u,v\right) =u+\frac{v}{u+\frac{\eta \alpha _{1}}{A_{1}}}+\delta ,
\end{equation*}%
where $\delta $ is given by $\left( 2.2\right) $. From $\left( 2.9\right) $
and $\left( 2.10\right) $, we obtain$\ \func{Re}\left\{ H\left(
q(z);zq^{^{\prime }}(z)\right) \right\} >0\ \left( z\in U\right) .$

\bigskip

To verify the condition%
\begin{equation}
\func{Re}\left\{ H\left( iu;t\right) \right\} \leq 0\ \ \ \ \left( u\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
;t\leq -\frac{1+u^{2}}{2}\right) ,  \tag{2.11}
\end{equation}%
we proceed as follows:%
\begin{eqnarray*}
\func{Re}\left\{ H\left( iu;t\right) \right\} &=&\func{Re}\left\{ iu+\frac{t%
}{iu+\frac{\eta \alpha _{1}}{A_{1}}}+\delta \right\} =\frac{\frac{\eta
\alpha _{1}}{A_{1}}t}{u^{2}+\left( \frac{\eta \alpha _{1}}{A_{1}}\right) ^{2}%
}+\delta \\
&\leq &-\frac{\Omega \left( \alpha _{1},A_{1},u,\delta \right) }{%
u^{2}+\left( \frac{\eta \alpha _{1}}{A_{1}}\right) ^{2}},
\end{eqnarray*}%
where%
\begin{equation}
\Omega \left( \alpha _{1},A_{1},u,\delta \right) =\left[ \frac{\eta \alpha
_{1}}{A_{1}}-2\delta \right] u^{2}-2\left( \frac{\eta \alpha _{1}}{A_{1}}%
\right) ^{2}\delta +\frac{\eta \alpha _{1}}{A_{1}}.  \tag{2.12}
\end{equation}%
For $\delta $\ given by $\left( 2.2\right) $, the coefficient of $u^{2}\ $in
the quadratic expression $\Omega \left( \alpha _{1},A_{1},u,\delta \right) $%
\ given by (2.12) is positive, which implies that (2,11) holds. Thus, by
using Lemma 2, we conclude that%
\begin{equation*}
\func{Re}\left\{ q\left( z\right) \right\} >0\ \ \ \ \left( z\in U\right) ,
\end{equation*}%
that is, that $G$ defined by (2.5) is convex (univalent) in $U$. To prove $%
F\prec G$, where $F$ and $G$ given by (2.5), let the function $L(z;t)$ be
defined by 
\begin{equation}
L\left( z,t\right) =G\left( z\right) +\frac{A_{1}\left( 1+t\right) }{\eta
\alpha _{1}}zG^{^{\prime }}\left( z\right) \ \ \ \left( 0\leq t<\infty ;z\in
U\right) .  \tag{2.13}
\end{equation}%
We note that 
\begin{equation*}
\left. \frac{\partial L\left( z,t\right) }{\partial z}\right\vert
_{z=0}=G^{^{\prime }}\left( 0\right) \left( \frac{A_{1}\left( 1+t\right)
+\eta \alpha _{1}}{\eta \alpha _{1}}\right) \neq 0\ \ \ \ \left( 0\leq
t<\infty ;z\in U\right) .
\end{equation*}%
This show that the function%
\begin{equation*}
L\left( z,t\right) =a_{1}\left( t\right) z+...\ ,
\end{equation*}%
satisfies the condition $a_{1}\left( t\right) \neq 0\ \left( 0\leq t<\infty
\right) .\ $Further, we have%
\begin{equation*}
\func{Re}\left\{ \frac{z\partial L\left( z,t\right) /\partial z}{\partial
L\left( z,t\right) /\partial t}\right\} =\func{Re}\left\{ \frac{\eta \alpha
_{1}}{A_{1}}+\left( 1+t\right) q\left( z\right) \right\} >0\ \ \ \left(
0\leq t<\infty ;z\in U\right) .
\end{equation*}%
Therefore, by using Lemma 1, $L\left( z,t\right) $ is a subordination chain.
It follows from the definition of subordination chain that%
\begin{equation*}
\phi \left( z\right) =G\left( z\right) +\frac{A_{1}}{\eta \alpha _{1}}%
zG^{^{\prime }}\left( z\right) =L\left( z,0\right)
\end{equation*}%
and%
\begin{equation*}
L\left( z,0\right) \prec L\left( z,t\right) \ \ \ \left( 0\leq t<\infty
\right) ,
\end{equation*}%
which implies that%
\begin{equation}
L\left( \zeta ,t\right) \notin L\left( U,0\right) =\phi \left( U\right) \ \
\ \left( 0\leq t<\infty ;\zeta \in \partial U\right) .  \tag{2.14}
\end{equation}%
If $F$ is not subordinate to $G$, by using Lemma 4, we know that there exist
two points $z_{0}\in U$ and $\zeta _{0}\in \partial U$ such that 
\begin{equation}
F\left( z_{0}\right) =G\left( \zeta _{0}\right) \text{\ \ \ and\ \ \ }%
z_{0}F^{^{\prime }}\left( z_{0}\right) =\left( 1+t\right) \zeta
_{0}G^{^{\prime }}\left( \zeta _{0}\right) \ \ \ \left( 0\leq t<\infty
\right) .  \tag{2.15}
\end{equation}%
Hence, by using $(2.5)$, $\left( 2.13\right) $,$\left( 2.15\right) \ $and
(2.3), we have%
\begin{equation*}
L\left( \zeta _{0},t\right) =G\left( \zeta _{0}\right) +\frac{A_{1}\left(
1+t\right) }{\eta \alpha _{1}}\zeta _{0}G^{^{\prime }}\left( \zeta
_{0}\right)
\end{equation*}

\begin{equation*}
=F\left( z_{0}\right) +\frac{A_{1}}{\eta \alpha _{1}}z_{0}F^{^{\prime
}}\left( z_{0}\right)
\end{equation*}

\begin{equation*}
=\left( \tfrac{\theta _{p,q,s}\left[ \alpha _{1}+1,A_{1},B_{1}\right]
f\left( z_{0}\right) }{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right]
f\left( z_{0}\right) }\right) \left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] f\left( z_{0}\right) }{z_{0}^{p}}\right) ^{\eta }\in
\phi \left( U\right) .
\end{equation*}%
This contradicts $\left( 2.14\right) $. Thus, we deduce that $F\prec G$.
Considering $F=G$, we see that the function $G$ is the best dominant. This
completes the proof of Theorem 1.\newline

We now derive the following superordination result.

\noindent \textbf{Theorem 2. }\textit{Let }$f,g\in A(p)\ $\textit{and }%
\begin{equation}
\func{Re}\left\{ 1+\frac{z\phi ^{^{\prime \prime }}\left( z\right) }{\phi
^{^{\prime }}\left( z\right) }\right\} >-\delta \ \ \ \ \left( \phi \left(
z\right) =\left( \tfrac{\theta _{p,q,s}\left[ \alpha _{1}+1,A_{1},B_{1}%
\right] g\left( z\right) }{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}%
\right] g\left( z\right) }\right) \left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] g\left( z\right) }{z^{p}}\right) ^{\eta }\right) , 
\tag{2.17}
\end{equation}%
\newline
\textit{where }$\delta $\textit{\ is given by }$\left( 2.2\right) .$\textit{%
\ If the\ function }$\left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1}+1,A_{1},B_{1}\right] f\left( z\right) }{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] f\left( z\right) }\right) \left( \tfrac{\theta
_{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right) }{z^{p}}%
\right) ^{\eta }$\textit{\ is univalent in }$U$\textit{\ and\ }$\left( 
\tfrac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right) 
}{z^{p}}\right) ^{\eta }\in \tciFourier ,$\textit{\ then the superordination
condition}%
\begin{equation}
\phi \left( z\right) \prec \left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1}+1,A_{1},B_{1}\right] f\left( z\right) }{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] f\left( z\right) }\right) \left( \tfrac{\theta
_{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right) }{z^{p}}%
\right) ^{\eta },  \tag{2.18}
\end{equation}%
\textit{implies that }%
\begin{equation}
\left( \frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] g\left(
z\right) }{z^{p}}\right) ^{\eta }\prec \left( \frac{\theta _{p,q,s}\left[
\alpha _{1},A_{1},B_{1}\right] f\left( z\right) }{z^{p}}\right) ^{\eta }, 
\tag{2.19}
\end{equation}%
\textit{where }$\left( \frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}%
\right] g\left( z\right) }{z^{p}}\right) ^{\eta }$\textit{\ is the best
subordinant.}

\noindent \textbf{Proof. }Suppose that the functions $F,G$ and $q$ are
defined by $\left( 2.5\right) $ and $\left( 2.6\right) $, respectively. By
applying similar method as in the proof of Theorem 1, we get%
\begin{equation*}
\func{Re}\left\{ q\left( z\right) \right\} >0\ \ \ \ \left( z\in U\right) .
\end{equation*}%
Next, to arrive at our desired result, we show that $G\prec F$. For this, we
suppose that the function $L\left( z,t\right) $ be defined by $\left(
2.13\right) $. Since $G$ is convex, by applying a similar method as in
Theorem 1, we deduce that $L\left( z,t\right) $ is subordination chain.
Therefore, by using Lemma 5, we conclude that $G\prec F$. Moreover, since
the differential equation%
\begin{equation*}
\phi \left( z\right) =G\left( z\right) +\frac{A_{1}}{\eta \alpha _{1}}%
zG^{^{\prime }}\left( z\right) =\varphi \left( G\left( z\right)
,zG^{^{\prime }}\left( z\right) \right)
\end{equation*}%
has a univalent solution $G$, it is the best subordinant. This completes the
proof of Theorem 2.\newline

Combining \ Theorem 1 and Theorem 2, we obtain the following "sandwich-type
result".

\noindent \textbf{Theorem 3.}\textit{\ Let }$f,g_{j}\in A(p)$\textit{\ and }%
\begin{equation*}
\func{Re}\left\{ 1+\frac{z\phi _{j}^{^{\prime \prime }}\left( z\right) }{%
\phi _{j}^{^{\prime }}\left( z\right) }\right\} >-\delta ,
\end{equation*}

\begin{equation*}
\left( \phi _{j}\left( z\right) =\left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1}+1,A_{1},B_{1}\right] g_{j}\left( z\right) }{\theta _{p,q,s}\left[
\alpha _{1},A_{1},B_{1}\right] g_{j}\left( z\right) }\right) \left( \tfrac{%
\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] g_{j}\left( z\right) }{%
z^{p}}\right) ^{\eta }\ \ \left( j=1,2\right) \right)
\end{equation*}%
\textit{where }$\delta $\textit{\ is given by }$\left( 2.2\right) .$\textit{%
\ If the function }$\left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1}+1,A_{1},B_{1}\right] f\left( z\right) }{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] f\left( z\right) }\right) \left( \tfrac{\theta
_{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right) }{z^{p}}%
\right) ^{\eta }$\textit{\ is univalent in }$U$\textit{\ and }$\left( \frac{%
\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right) }{z^{p}%
}\right) ^{\eta }\in \tciFourier ,$\textit{\ then the condition}%
\begin{equation}
\phi _{1}\left( z\right) \prec \left( \tfrac{\theta _{p,q,s}\left[ \alpha
_{1}+1,A_{1},B_{1}\right] f\left( z\right) }{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] f\left( z\right) }\right) \left( \tfrac{\theta
_{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right) }{z^{p}}%
\right) ^{\eta }\prec \phi _{2}\left( z\right) ,  \tag{2.20}
\end{equation}%
\textit{implies that}%
\begin{equation}
\left( \frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right]
g_{1}\left( z\right) }{z^{p}}\right) ^{\eta }\prec \left( \frac{\theta
_{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] f\left( z\right) }{z^{p}}%
\right) ^{\eta }\prec \left( \frac{\theta _{p,q,s}\left[ \alpha
_{1},A_{1},B_{1}\right] g_{2}\left( z\right) }{z^{p}}\right) ^{\eta }, 
\tag{2.21}
\end{equation}%
\textit{where }$\left( \frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}%
\right] g_{1}\left( z\right) }{z^{p}}\right) ^{\eta }$\textit{\ and }$\left( 
\frac{\theta _{p,q,s}\left[ \alpha _{1},A_{1},B_{1}\right] g_{2}\left(
z\right) }{z^{p}}\right) ^{\eta }$\textit{\ are , respectively, the best
subordinant and the best dominant.}\bigskip

\noindent \textbf{Remark 1.} \textit{Specializing }$\ q$\textit{, }$s,\alpha
_{1},A_{1},...,\alpha _{q},A_{q}$\textit{\ and }$\beta _{1},B_{1},...\beta
_{s},B_{s},$\textit{\ in the above results, we obtain the corresponding
results for different classes associated with\ the operators (1-7) defined
in the introduction.}\newline

\noindent {\LARGE References}

\noindent \lbrack 1] H. A. Al-Kharasani and S.S. Al-Hajiry, A linear
operator and its applications on p-valent functions, Internat. J. Math.
Analysis, (2007), 627-634.

\noindent \lbrack 2] M. K. Aouf and J. Dziok, Certain class of analytic
functions associated with the Wright generalized hypergeometric function, J.
Math. Appl. 30(2008), 23-32.

\noindent \lbrack 3] M. K. Aouf, A. Shamandy, A. O. Mostafa and S. M.
Madian, Certain class of p-valent functions associated with the Wright
generalized hypergeometric function, Demonstratio Math., (2010), no. 1,
40-54.

\noindent \lbrack 4] N. E. Cho, O.S. Kwon and H.M. Srivastava, Inclusion and
argument properties for certain subclasses of multivalent functions
associated with a family of linear operators, J. Math. Anal. Appl., 292
(2004), 470--483.

\noindent \lbrack 5] J. H. Choi, M. Saigo and H.M. Srivastava, Some
inclusion properties of a certain family of integral operators, J. Math.
Anal. Appl., 276 (2002), no.1, 432--445.

\noindent \lbrack 6] J. Dziok and R. K. Raina, Families of analytic
functions associated with the Wright generalized hypergeometric function,
Demonstratio Math., 37(2004), no.3, 533-542.

\noindent \lbrack 7] J. Dziok and H. M. Srivastava, Classes of analytic
functions associated with the generalized hypergeometric function, Appl.
Math. Comput. 103 (1999), 1--13.

\noindent \lbrack 8] R.M. Goel and N.S. Sohi, A new criterion for p-valent
functions, Proc. Amer. Math. Soc., 78 (1980), 353--357.

\noindent \lbrack 9] J.-L. Liu and K.I. Noor, Some properties of Noor
integral operator, J. Natur. Geom., 21 (2002), 81--90.

\noindent \lbrack 10] S. S. Miller and P. T. Mocanu, Differential
subordinations and univalent functions, Michigan Math. J. 28 (1981), no. 2,
157--172.

\noindent \lbrack 11] S. S. Miller and P. T. Mocanu, Univalent solutions of
Briot-Bouquet differential equations, J. Differential Equations 56 (1985),
no. 3, 297--309.

\noindent \lbrack 12] S. S. Miller and P. T. Mocanu, Differential
Subordinations: Theory and Applications, Series on Monographs and Textbooks
in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York and
Basel, 2000.

\noindent \lbrack 13] S. Miller and P. T. Mocanu, Subordinants of
differential superordinations, Complex Var. Theory Appl. 48(2003), no.10,
815--826.

\noindent \lbrack 14] J. Patel and A.K. Mishra, On certain subclasses of
multivalent functions associated with an extended fractional differintegral
operator, J. Math. Anal. Appl., 332 (2007), 109--122.

\noindent \lbrack 15] C. Pommerenke, Univalent Functions, Vandenhoeck and
Ruprecht, G$\overset{..}{o}$ttingen, 1975.

\noindent \lbrack 16] R. K. Raina, On certain classes of analytic functions
and application to fractional calculas operator, Integral Transform. Spec.
Funct. 5(1997), 247-260.

\noindent \lbrack 17] R. K. Raina and T. S. Nahar, On univalent and starlike
Wright generalized hypergeometric functions, Rend. Sen. Mat. Univ. Padova
95(1996), 11-22

\noindent \lbrack 18] H. Saitoh, A linear operator and its applications of
first order differential subordinations, Math. Japon., 44 (1996), 31--38.

\noindent \lbrack 19] H. M. Srivastava and M.K. Aouf, A certain fractional
derivative operator and its applications to a new class of analytic and
multivalent functions with negative coefficients. I, J. Math. Anal. Appl.,
171 (1992), 1-13; II, J. Math. Anal. Appl., 192 (1995), 673-688.

\noindent \lbrack 20] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian
Hypergeometric Series, Ellis Horwood Ltd., Chichester, Halsted Press (John
Wiley \& Sons, Inc.), New York, 1985.

\noindent \lbrack 21] E. M. Wright, The asymptotic expansion of the
generalized hypergeometric functions, Proc. London Math. Soc. 46(1946),
389-408.

\end{document}