Abstract and Applied Analysis

Volume 2014 (2014), Article ID 346162, 4 pages

http://dx.doi.org/10.1155/2014/346162

Research Article

On a New Criterion for Meromorphic Starlike Functions

Lei Shi and Zhi-Gang Wang

School of Mathematics and Statistics, Anyang Normal University, Anyang Henan 455000, China

Received 14 December 2013; Accepted 17 February 2014; Published 19 March 2014

Academic Editor: Mohamed K. Aouf

Copyright © 2014 Lei Shi and Zhi-Gang Wang. 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.

The main purpose of this paper is to derive a new criterion for meromorphic starlike functions of order α.

1. Introduction and Preliminaries

Let denote the class of functions of the form

which are analytic in the punctured open unit disk

A function is said to be in the class of meromorphic starlike functions of order if it satisfies the condition

For simplicity, we write .

For two functions and , analytic in , we say that the function is subordinate to in and write

if there exists a Schwarz function , which is analytic in with

such that

Indeed, it is known that

Furthermore, if the function is univalent in , then we have the following equivalence:

In a recent paper, Miller et al. [1] proved the following result.

Theorem A. Let , , and

If satisfies the condition

then .

More recently, Catas [2] improved Theorem A as follows.

Theorem B. Let , , and

where is given by (9) and

If satisfies the condition

then .

In this paper, we aim at finding the conditions for starlikeness of the expression for .

For some recent investigations of meromorphic functions, see, for example, the works of [3–12] and the references cited therein.

In order to prove our main results, we require the following subordination result due to Hallenbeck and Ruscheweyh [13].

Lemma 1. Let be a convex function with , and let be a complex number with . If a function

satisfies the condition

then

2. Main Results

We begin by stating the following result.

Theorem 2. Let , , and . If satisfies the inequality

where

then .

Proof. Suppose that

It follows from (19) that

By combining (17), (19), and (20), we easily get

or equivalently

An application of Lemma 1 yields

The subordination (23) is equivalent to

From (18) and (24), we know that

We suppose that

By virtue of (19) and (26), we get

which implies that (17) can be written as

We now only need to show that (28) implies in . Indeed, if this is false, since , then there exists a point such that , where is a real number. Thus, in order to show that (28) implies in , it suffices to obtain the contradiction from the inequality

By setting

we have

By means of (24), we obtain

It follows from (31) and (32) that

We now set

If , then (29) holds true. Since , the inequality holds if the discriminant ; that is,

and the last inequality is equivalent to

Furthermore, in view of (24) and (36), after a geometric argument, we deduce that

It follows from (37) that , which implies that . But this contradicts (28). Therefore, we know that in . By virtue of (26), we conclude that

This evidently completes the proof of Theorem 2.

Taking in Theorem 2, we obtain the following result.

Corollary 3. Let and . If satisfies the inequality

then .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grants nos. 11301008, 11226088, and 11101053, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant no. 14B110012 of China.

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