Research Article | Open Access

# On a New Criterion for Meromorphic Starlike Functions

**Academic Editor:**Mohamed K. Aouf

#### 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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#### Copyright

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.