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Journal of Applied Mathematics
Volume 2011, Article ID 804150, 8 pages
http://dx.doi.org/10.1155/2011/804150
Research Article

Starlikeness Properties of a New Integral Operator for Meromorphic Functions

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D. Ehsan, Malaysia

Received 3 March 2011; Revised 15 April 2011; Accepted 3 June 2011

Academic Editor: Pablo Gonza'lez-Vera

Copyright © 2011 Aabed Mohammed and Maslina Darus. 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

We define here an integral operator for meromorphic functions in the punctured open unit disk. Several starlikeness conditions for the integral operator are derived.

1. Introduction

Let denotes the class of functions of the form which are analytic in the punctured open unit disk where is the open unit disk .

We say that a function is meromorphic starlike of order , and belongs to the class , if it satisfies the inequality

A function is a meromorphic convex function of order , if satisfies the following inequality and we denote this class by .

Analogous to the integral operator defined by Breaz et al. [1] on the normalized analytic functions, we now define the following integral operator on the space meromorphic functions in the class .

Definition 1.1. Let . We define the integral operator by For the sake of simplicity, from now on we will write instead of .
By , we denote the class of functions such that In order to derive our main results, we have to recall here the following preliminary results.

Lemma 1.2 (see [2]). Suppose that the function satisfies the following condition: If the function is analytic in and then,

Proposition 1.3 (see [3]). If satisfying then,

2. Starlikeness of the Operator

In this section, we investigate sufficient conditions for the integral operator which is defined in Definition 1.1, to be in the class .

Theorem 2.1. Let for all. If then belongs to .

Proof. On successive differentiation of , which is defined in (1.5), we get Then from (2.2), we obtain By multiplying (2.3) with yield, That is equivalent to Or We can write the left-hand side of (2.6), as the following: We define the regular function in by and . Differentiating logarithmically, we obtain From (2.7), (2.8), and (2.9), we obtain Let us put From (2.1), (2.10), and (2.11), we obtain Now, we proceed to show that Indeed, from (2.11), we have Thus, from (2.12), (2.14), and by using Lemma 1.2, we conclude that , and so that is, is starlike of order 0.

Theorem 2.2. For , let and . If be the integral operator given by (1.5) and Then belong to , where .

Proof. Following the same steps as in Theorem 2.1, we obtain Taking the real part of both terms of the last expression, we have Since for , we receive Therefore, Using (2.16), (2.20), and applying Proposition 1.3, we get , where .

Letting in Theorem 2.2, we get the following.

Corollary 2.3. For , let and . If be the integral operator given by (1.5) and Then is starlike of order .

Theorem 2.4. For , let and . If be the integral operator given by (1.5) and Then is starlike of order .

Proof. Following the same steps as in Theorem 2.1, we obtain We calculate the real part from both terms of the above equality and obtain Since for all , the above relation then yields Because , we obtain that Using (2.24), (2.28) and applying Proposition 1.3, we get is a starlike function of order

Letting in Theorem 2.4, we get the following.

Corollary 2.5. For , let and . If   be the integral operator given by (1.5) and Then is starlike of order .

Letting and in Corollary 2.5, we get the following.

Corollary 2.6. Let , and . If be the integral operator, Then is starlike of order .

Other work related to integral operator for different studies can also be found in [4–6].

Acknowledgments

The work here was supported by MOHE Grant: UKM-ST-06-FRGS0244-2010. The authors also would like to thank the referee for his/her careful reading and making some valuable comments which have improved the presentation of this paper.

References

  1. D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” Acta Universitatis Apulensis, no. 16, pp. 11–16, 2008. View at Google Scholar · View at Zentralblatt MATH
  2. S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Mohammed and M. Darus, “New criteria for meromorphic functions,” Submitted.
  4. A. Mohammed and M. Darus, “New properties for certain integral operators,” International Journal of Mathematical Analysis, vol. 4, no. 41–44, pp. 2101–2109, 2010. View at Google Scholar
  5. A. Mohammed and M. Darus, “A new integral operator for meromorphic functions,” Acta Universitatis Apulensis, no. 24, pp. 231–238, 2010. View at Google Scholar
  6. N. Breaz, D. Breaz, and M. Darus, “Convexity properties for some general integral operators on uniformly analytic functions classes,” Computers and Mathematics with Applications, vol. 60, no. 12, pp. 3105–3107, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH