Research Article | Open Access

# Starlikeness Properties of a New Integral Operator for Meromorphic Functions

**Academic Editor:**Pablo Gonza'lez-Vera

#### 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

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#### Copyright

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.