The Scientific World Journal

Volume 2014 (2014), Article ID 806168, 6 pages

http://dx.doi.org/10.1155/2014/806168

Research Article

On a Subclass of Meromorphic Close-to-Convex Functions

Ming-Liang Li,1,2 Lei Shi,³ and Zhi-Gang Wang³

¹School of Mathematics and Statistics, Central South University, Changsha, Hunan 410075, China

²School of Mathematics and Computing Science, Hunan First Normal University, Changsha, Hunan 410205, China

³School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China

Received 5 July 2014; Accepted 14 August 2014; Published 28 August 2014

Academic Editor: Rosihan M. Ali

Copyright © 2014 Ming-Liang Li et al. 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 introduce and investigate a certain subclass of meromorphic close-to-convex functions. Such results as coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity are derived.

1. Introduction and Preliminaries

Let denote the class of functions of the form

which are analytic in the punctured open unit disk

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:

Let , where is given by (1) and is defined by

Then the Hadamard product (or convolution) of the functions and is defined by

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

A function is said to be in the class of meromorphic convex functions if it satisfies the inequality

Moreover, a function is said to be in the class of meromorphic close-to-convex functions if it satisfies the condition

Let

be analytic in . If there exists a function such that

then we say that , where denotes the usual class of convex functions. The function class was introduced and studied recently by Peng [1] (see also Peng and Han [2], Selvaraj [3], Gao and Zhou [4], Kowalczyk and Leś-Bomba [5], and Xu et al. [6]).

Motivated essentially by the above mentioned function class , we now introduce and investigate the following class of meromorphic close-to-convex functions.

Definition 1. A function is said to be in the class if it satisfies the inequality

where (and throughout this paper unless otherwise mentioned) the parameters and are constrained as follows:

It is easy to verify that if and only if

We observe that

and, thus, the function class is a subclass of meromorphic close-to-convex functions.

Clearly, the class is the familiar class of meromorphic close-to-convex functions of order .

For some recent investigations of meromorphic functions, see, for example, the works of [7–22] and the references cited therein.

To derive our main results, we need the following lemmas.

Lemma 2 (see [23]). Let

be analytic in and let

be analytic and convex in . If , then

Lemma 3 (see [24]). Suppose that

Then

Each of these inequalities is sharp, with the extremal function given by

Lemma 4 (see [25]). Let and . Then,

if and only if

Lemma 5 (see [26]). Suppose that the function . Then

Lemma 6 (see [27]). Suppose that

Then,

where is the unique root of the equation

in the interval . The results are sharp.

In the present paper, we aim at proving some coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity of the class .

2. Main Results

We begin by stating the following coefficient inequality of the class .

Theorem 7. Suppose that

Then

Proof. Let and suppose that

where

It follows that

In view of Lemma 2, we know that

By substituting the series expressions of functions , , and into (34), we get

Since is univalent in , it is well known that .

On the other hand, we find from (38) that

By noting that , it follows from Lemma 3 that

Combining (37), (39), and (40), we have

Thus, the assertion (33) of Theorem 7 follows directly from (41).

Theorem 8. Let

If satisfies the condition

then .

Proof. To prove , it suffices to show that (15) holds. From (43), we know that

Now, by the maximum modulus principle, we deduce from (1) and (44) that

This evidently completes the proof of Theorem 8.

Example 9. By applying Theorem 8, it is obvious to see that the function

Theorem 10. Let and . A function if and only if

Proof. A function if and only if

It is easy to see that condition (48) can be written as

We observe that

By substituting (50) into (49), we get the desired assertion (47) of Theorem 10.

Theorem 11. Let

Then,

Proof. Suppose that . We easily know that

By setting , , , and , it follows from (51) that

In view of Lemma 4, we deduce that

which implies that . Thus, the assertion (52) of Theorem 11 holds.

Theorem 12. Let . Then,

Proof. Let . By definition, we know that

Suppose that the function is defined by (36). Then, we have

Since , by Lemma 5, we know that

Thus, by virtue of (36), (58), and (59), we readily get the assertion (56) of Theorem 12.

Finally, we derive the radius of meromorphic convexity for the class .

Theorem 13. Let with . Then,

(1)for , is meromorphic convex in ;(2)for , is meromorphic convex in ,

where is the unique root of the equation

in the interval and and are the smallest root of the equations

in the interval , respectively.

Proof. Let and suppose that

Then,

It follows from (62) that

Differentiating both sides of (64) logarithmically, we get

Since , we know that

Combining (63), (65), (66), and Lemma 6, we obtain

where is the unique root of (30) in the interval . It follows from (67) that the bound of meromorphic convexity for the class is determined either by the equation

or by the equation

We note that (68) and (69) can be rewritten as follows:

Let and be the smallest root of the equations and in the interval , respectively. By observing that , we deduce that for .

Similarly, we know that for , since .

We observe that

Thus, when , is meromorphic convex in ; when , is meromorphic convex in .

The proof of Theorem 13 is thus completed.

Conflict of Interests

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

Acknowledgments

This research was supported by the National Natural Science Foundation under Grant nos. 11301008 and 11226088, 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 the People's Republic of China. The authors would like to thank the referees for their valuable comments and suggestions which essentially improved the quality of this paper.

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