Research Article | Open Access

# Differential Subordinations for Certain Meromorphically Multivalent Functions Defined by Dziok-Srivastava Operator

**Academic Editor:**Malisa R. Zizovic

#### Abstract

By making use of the Dziok-Srivastava operator, we introduce a new class of meromorphically multivalent functions. Some inclusion properties of functions belonging to this class are derived.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the punctured open unit disk with a pole at . Also let the Hadamard product (or convolution) of the following functions: be given by

Given two functions and , which are analytic in , we say that the function is subordinate to and write or (more precisely) , if there exists a Schwarz function , analytic in with and such that . In particular, if is univalent in , we have the following equivalence:

Let be the class of functions of the form which are analytic in . A function is said to be in the class if for some . When , is the class of starlike functions of order in . A function is said to be prestarlike of order in if where the symbol means the familiar Hadamard product (or convolution) of two analytic functions in . We denote this class by (see [1]). Clearly a function is in the class if and only if is convex univalent in and .

For complex parameters we define the generalized hypergeometric function by where is the Pochhammer symbol defined, in terms of the Gamma function ,by Corresponding to a function defined by we now consider a linear operator defined by means of the Hadamard product (or convolution) as follows: For convenience, we write Thus, after some calculations, we have

The operator is popularly known as the generalized Dziok-Srivastava operator. Many interesting subclasses of multivalent functions, associated with the operator and its various special cases, were investigated recently by (e.g.) Dziok and Srivastava [2–4], Liu [5], Liu and Srivastava [6, 7], Patel et al. [8], Wang et al. [9], and others.

Let be the class of functions with , which are analytic and convex univalent in .

*Definition 1.1. *A function is said to be in the class if it satisfies the subordination condition
where is a complex number and .

The main object of this paper is to present a systematic investigation of the class defined above by means of the generalized Dziok-Srivastava operator .

For our purpose, we shall need the following lemmas to derive our main results for the class .

Lemma 1.2 (see [10]). *Let be analytic in and be analytic and convex univalent in with . If
**
where , then
**
and is the best dominant of (1.17).*

Lemma 1.3 (see [1]). *Let , and . Then, for any analytic function in ,
**
where denotes the closed convex hull of .*

#### 2. Properties of the Class

Theorem 2.1. *Let . Then .*

*Proof. *Let and suppose that
for . Then the function is analytic in with . Differentiating both sides of (2.1) with respect to and using (1.16), we have
Hence an application of Lemma 1.2 yields
Noting that and that is convex univalent in , it follows from (2.1) to (2.3) that
Thus and the proof of Theorem 2.1 is completed.

Theorem 2.2. *Let . Then .*

*Proof. *Define a function by
Then
where
is defined as in (1.11), and
By (2.8), we see that
which implies that

Let . It is easy to verify that
From (2.11), (2.12), and (2.6), we deduce that
where
Since the function belongs to the function class and is convex univalent in , it follows from (2.12), (2.13), (2.14), and Lemma 1.3 that
Thus and the proof of Theorem 2.2 is completed.

Theorem 2.3. *Let and
**
Then
*

*Proof. *For and , we have
where
In view of (2.16), the function has the Herglotz representation
where is a probability measure defined on the unit circle and
Since is convex univalent in , it follows from (2.18) to (2.20) that
This shows that and the theorem is proved.

Theorem 2.4. *Let and
**
Then
*

*Proof. *For and , from (2.18) we have
where is defined as in (2.19).

Since is convex univalent in ,
it follows from (2.25) and Lemma 1.3 the desired result.

Theorem 2.5. *Let and . If , where
**
then . The bound is sharp when .*

*Proof. *Let us define
for with and . Then we have
Hence an application of Lemma 1.2 yields
where

If , where is given by (2.27), then it follows from (2.31) that
Now, by using the Herglotz representation for , from (2.28) and (2.30), we arrive at
because is convex univalent in . This shows that .

For and defined by
it is easy to verify that
Thus . Also, for , we have
which implies that . Hence the bound cannot be increased when .

#### Acknowledgment

The authers would like to express sincere thanks to the referees for careful reading and suggestions which helped us to improve the paper.

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#### Copyright

Copyright © 2011 Ying Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.