Journal of Complex AnalysisVolume 2013 (2013), Article ID 690584, 4 pageshttp://dx.doi.org/10.1155/2013/690584

Research Article

Starlikeness of General Integral Operator for Meromorphic Multivalent Functions

Pranay Goswami¹ and Serap Bulut²

¹School of Liberal Studies, Bharat Ratna Dr B.R. Ambedkar University, Delhi 110006, India²Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Kocaeli, Turkey

Received 31 August 2012; Accepted 30 November 2012

Academic Editor: Jacek Dziok

Copyright © 2013 Pranay Goswami and Serap Bulut. 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 introduce a new integral operator for meromorphic multivalent functions. The starlikeness condition of this integral operator is determined. Several special cases are also discussed in the form of corollaries.

1. Introduction

Let denote the class of all meromorphic functions of the form which are analytic and -valent in the punctured open unit disk where is the open unit disk . In particular, we set .

A function is said to be meromorphic -valent starlike and belongs to the class , if it satisfies the inequality:

A function is said to be meromorphic -valent convex and belongs to the class , if it satisfies the inequality:

We note that In particular, we set for the classes of meromorphic starlike and convex functions, respectively.

Analogous to the integral operator defined by Frasin [1] on normalized analytic functions, we now define the following general integral operator for the meromorphic -valent functions in the class .

Definition 1. Let , for all and . We introduce a new general integral operator

Remark 2. The integral operator introduced here generalizes the following integral operators.(i)If , then we have the integral operator introduced and studied by Bulut and Goswami [2].(ii)If , then it reduces to the integral operator (iii)If , then it reduces to the integral operator (iv)For , , , and , it reduces to the new integral operator

The integral operators defined in (ii) and (iii) are introduced and studied by Mohammed and Darus [3] (also the case in (ii) and (iii) is defined in [4] and [12], resp.). Furthermore, the case for the integral operator in (iv) is defined by Bulut and Goswami [2]. For the recent developments on meromorphic functions, refer to [5–10].

For the starlikeness of the integral operator defined in Definition 1, we need to use the following lemma.

Lemma 3 (see [11]). Let satisfy the following condition: If the function , where , is analytic in and then

2. Starlikeness of the Integral Operator

In this section, we investigate sufficient conditions for the meromorphically starlikeness of the integral operator which is defined in Definition 1.

Theorem 4. Let ,  for all and . If for all , then the general integral operator defined in Definition 1 belongs to the meromorphic starlike function class .

Proof. For the sake of simplicity, in the proof, we shall write instead of . From (7), it is easy to see that Differentiating the above equality with respect to , we obtain From (16) and (17), we get or equivalently After some calculations, we have We can write the left-hand side of (20) as the following: Now we define a regular function by and . Differentiating (22) logarithmically, we obtain From (21), (22), and (23), we have Let us put From (15), (24) and (25), we get Now we have to show that From (25), we have Thus using Lemma 3, and from (26) and (28), we conclude that , and so that is, the integral operator is a meromorphic multivalent starlike function.

Putting in Theorem 4, we get the following Corollary.

Corollary 5 (see [2, Theorem 2.1]). Let , for all and . If for all , then the general integral operator defined in (8) belongs to the meromorphically starlike function class .

Taking in Theorem 4, we get the following.

Corollary 6. Let , for all and . If for all , then the general integral operator defined in (9) belongs to the meromorphically starlike function class .

Remark 7. If we set in Corollary 6, then we have [12, Theorem 2.1].

Next, putting in Theorem 4, we get the following.

Corollary 8. Let , for all and . If for all , then the general integral operator defined in (10) belongs to the meromorphically starlike function class .

Further taking , , , and in Theorem 4, we have the following.

Corollary 9. Let and . If then the general integral operator defined in (11) belongs to the meromorphically starlike function class .

References

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