Research Article | Open Access
On a New Criterion for Meromorphic Starlike Functions
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.  proved the following result.
Theorem A. Let , , and If satisfies the condition then .
More recently, Catas  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 .
In order to prove our main results, we require the following subordination result due to Hallenbeck and Ruscheweyh .
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
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.
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 © 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.