Journal of Complex Analysis

Volume 2013 (2013), Article ID 676027, 5 pages

http://dx.doi.org/10.1155/2013/676027

## Convolution Properties of -Valent Functions Associated with a Generalization of the Srivastava-Attiya Operator

^{1}Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa^{2}Department of Mathematics, Sambalpur University, Jyoti Vihar, Burla, Odisha 768019, India

Received 15 August 2012; Accepted 4 November 2012

Academic Editor: Jacek Dziok

Copyright © 2013 Priyabrat Gochhayat. 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.

#### Abstract

Let denote the class of functions analytic in the open unit disc and given by the series . For , the transformation defined by , has been recently studied as *fractional differintegral operator* by Mishra and Gochhayat (2010). In the present paper, we observed that can also be viewed as a generalization of the Srivastava-Attiya operator. Convolution preserving properties for a class of multivalent analytic functions involving an adaptation of the popular Srivastava-Attiya transform are investigated.

#### 1. Introduction and Preliminaries

Let be the class of functions analytic in the *open* unit disk

Suppose that and are in . We say that is *subordinate* to (or is *superordinate* to ), written as
if there exists a function , satisfying the conditions of the Schwarz lemma and such that
It follows that
In particular, if is *univalent* in , then the reverse implication also holds (cf. [1]).

For real parameters and such that , recalling the function of the form: which maps conformally onto a disk (whenever ), symmetrical with respect to the real axis, which is centered at the point and with its radius equal to Furthermore, the boundary circle of the disk intersects the real axis at the point and provided .

In this paper we will also be dealing with the subclass of consisting of functions of the following form:

With a view to define the Srivastava-Attiya transform we recall here a general Hurwitz-Lerch-Zeta function, which is defined in [2, 3] by the following series:

Important special cases of the function include, for example, the Reimann zeta function , the Hurwitz zeta function , the Lerch zeta function , the polylogarithm and so on. Recent results on , can be found in the expositions [4, 5].

By making use of the following normalized function: Srivastava-Attiya [2] introduced the linear operator by the following series: where the function is, respectively, by The operator is now popularly known in the literature as the Srivastava-Attiya operator. Various basic properties of are systematically investigated in [6–11].

For a function and represented by the series (8), the transformation
defined by
has been recently studied as *fractional differintegral operator* by the authors [12]. We observed that can also be viewed as a generalization of the Srivastava-Attiya operator (take in (14)), suitable for the study of multivalent functions. (Also see [13] for a variant.)

Furthermore, transformation generalizes several previously studied familiar operators. For example taking we get the identity transformation; the choices yield the Alexander transformation and a negative integer, give the Sălăgean operator. Some more interesting particular cases are also pointed out by the authors in [12] (also see [14]).

Using (14) it can be verified that For the functions given by their Hadamard product (or convolution) is defined by Observe that when , the operator given by (14) can be represented in terms of convolution as follows: where In the sequel to earlier investigations, in the present paper we find a convolution result involving is also presented.

With a view to state a well-known result, we denote by the class of functions as follows: which are analytic in and satisfy . A function if and only if . The following result is a consequence of the principle of subordination.

Lemma 1. *Let the function , given by (20), be in the class . Then
*

#### 2. Convolution Properties of

We state and prove the following convolution preserving properties of .

Theorem 2. *Let and . If each of the functions satisfies the following subordination condition:
**
then
**
where
**
The result is the best possible for .*

*Proof. *Suppose that each of the functions satisfies the condition (22). Set

Then, by making use of the identity (15) in (26) we get
Therefore, a simple computation, by using (24) and (27), shows that
where
The proof will be completed by finding the best possible lower bound for . A change of variable also gives

Since , where , it follows from a result in [15] that
and the bound is the best possible. An application of Lemma 1, in (30), yields
In order to show that is the best possible in the assertion (23) when , we consider the function given by
It is readily checked that satisfies (22) with . Since
it follows from (30) and Lemma 1 that
This completes the proof of Theorem 2.

Taking we get the following consequence.

Corollary 3. *Let and . If each of the functions satisfies the following subordination condition:
**
then
**
where
**
The result is best possible for .*

*Remark 4. *Taking in Corollary 3, we get the result due to Özkan (cf. [16, Theorem 1]).

#### Acknowledgment

The present investigation is partially supported under the University of Pretoria, Post Doctoral Fellowship Grant and the Fast Track Research Project for Young Scientiest, Department of Science and Technology, Government of India, sanction Letter no. 100/IFD/12100/2010-11.

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