/ / Article

Research Article | Open Access

Volume 2013 |Article ID 676027 | 5 pages | https://doi.org/10.1155/2013/676027

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

Accepted04 Nov 2012
Published07 Feb 2013

#### 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. ).

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  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 .

For a function and represented by the series (8), the transformation defined by has been recently studied as fractional differintegral operator by the authors . 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  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  (also see ).

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  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.

1. S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics No. 225, Marcel Dekker, New York, NY, USA, 2000. View at: MathSciNet
2. H. M. Srivastava and A. A. Attiya, “An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination,” Integral Transforms and Special Functions, vol. 18, no. 3, pp. 207–216, 2007.
3. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, London, UK, 2001. View at: MathSciNet
4. H. M. Srivastava, D. Jankov, T. K. Pogány, and R. K. Saxena, “Two-sided inequalities for the extended Hurwitz-Lerch Zeta function,” Computers and Mathematics with Applications, vol. 62, no. 1, pp. 516–522, 2011.
5. H. M. Srivastava, R. K. Saxena, T. K. Pogány, and R. Saxena, “Integral and computational representations of the extended Hurwitz-Lerch zeta function,” Integral Transforms and Special Functions, vol. 22, no. 7, pp. 487–506, 2011.
6. M. K. Aouf, A. O. Mostafa, A. Shahin, and S. Madian, “Applications of differential subordinations for certain class of p-valent functions associated with generalized Srivastava-Attiya operator,” Journal of Inequalities and Applications, vol. 2012, article 153, 2012. View at: Publisher Site | Google Scholar | MathSciNet
7. N. E. Cho, I. H. Kim, and H. M. Srivastava, “Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 918–928, 2010.
8. G. Murugusundaramoorthy, “Subordination results for spiral-like functions associated with Srivastva-Attiya operator,” Integral Transforms and Special Functions, vol. 23, no. 2, pp. 97–103, 2012.
9. K. I. Noor and S. Z. H. Bukhari, “Some subclasses of analytic and spiral-like functions of complex order involving the Srivastava-Attiya integral operator,” Integral Transforms and Special Functions, vol. 21, no. 12, pp. 907–916, 2010.
10. S. M. yuan and Z. M. liu, “Some properties of two subclasses of k-fold symmetric functions associated with Srivastava-Attiya operator,” Applied Mathematics and Computation, vol. 218, pp. 1136–1141, 2011. View at: Publisher Site | Google Scholar | MathSciNet
11. Z. G. Wang, Q. G. Li, and Y. P. Jiang, “Certain subclasses of multivalent analytic functions involving the generalized Srivastava-Attiya operator,” Integral Transforms and Special Functions, vol. 21, no. 3, pp. 221–234, 2010.
12. A. K. Mishra and P. Gochhayat, “Invariance of some subclass of multivalent functions under a differintegral operator,” Complex Variables and Elliptic Equations, vol. 55, no. 7, pp. 677–689, 2010.
13. J. L. Liu, “Sufficient conditions for strongly star-like functions involving the generalized Srivastava-attiya operator,” Integral Transforms and Special Functions, vol. 22, no. 2, pp. 79–90, 2011.
14. A. O. Mostafa and M. K. Aouf, “Some applications of differential subordination of p-valent functions associated with Cho-Kwon-Srivastava operator,” Acta Mathematica Sinica, vol. 25, no. 9, pp. 1483–1496, 2009. View at: Publisher Site | Google Scholar | MathSciNet
15. J. Stankiewicz and Z. Stankiewicz, “Some applications of Hadamard convolution in the theory of functions,” Annales Universitatis Mariae Curie-Sklodowska Lodowska A, vol. 40, no. 2, pp. 251–265, 1986. View at: Google Scholar | MathSciNet
16. Ö. Özkan, “Some subordination results of multivalent functions defined by integral operator,” Journal of Inequalities and Applications, vol. 2007, Article ID 71616, 2007.