Research Article | Open Access
Differential Subordinations for Certain Meromorphically Multivalent Functions Defined by Dziok-Srivastava Operator
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.
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 ). 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 , Liu and Srivastava [6, 7], Patel et al. , Wang et al. , 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 1.3 (see ). 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
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
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
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 .
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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