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.