Journal of Applied Mathematics

Journal of Applied Mathematics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 804150 | 8 pages | https://doi.org/10.1155/2011/804150

Starlikeness Properties of a New Integral Operator for Meromorphic Functions

Academic Editor: Pablo Gonza'lez-Vera
Received03 Mar 2011
Revised15 Apr 2011
Accepted03 Jun 2011
Published26 Jul 2011

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

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.


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