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Journal of Complex Analysis

Volume 2013 (2013), Article ID 704784, 5 pages

http://dx.doi.org/10.1155/2013/704784

## On Certain New Subclass of Meromorphic Close-to-Convex Functions

^{1}School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, China^{2}School of Railway Tracks and Transportation, East China Jiao Tong University, Nanchang, Jiangxi 330013, China

Received 19 November 2012; Accepted 31 January 2013

Academic Editor: Mikael Lindstrom

Copyright © 2013 Jing-Ping Yi 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

We introduce a certain new subclass of meromorphic close-to-convex functions. Such results as inclusion relationship, coefficient inequalities, distortion, and growth theorems for this class of functions are derived.

#### 1. Introduction

Let denote the class of functions of the form
which are *analytic* in the *punctured* open unit disk:

Let denote the class of functions given by which are analytic and convex in and satisfy the following condition:

A function is said to be in the class of *meromorphic starlike functions of order * if it satisfies the following inequality:
Moreover, a function is said to be in the class of *meromorphic close-to-convex functions* if it satisfies the following condition:

For two functions and analytic in , we say that the function is subordinate to in and write , if there exists a Schwarz function , analytic in with and such that . Indeed, it is well known that Furthermore, if the function is univalent in , then we have the following equivalence:

Recently, Wang et al. [1] introduced and investigated the class of meromorphic close-to-convex functions which satisfy the inequality where . Kowalczyk and Leś-Bomba [2] discussed the class of analytic functions related to the starlike functions; a function which is analytic in is said to be in the class , if it is satisfies the following inequality: where . Şeker [3] discussed the class of analytic functions which satisfy the following condition: where , and .

Motivated essentially by the classes , and , we introduce and study the following more generalized class of meromorphic functions.

*Definition 1. *A function is said to be in the class if it satisfies the following inequality:
where is a fixed positive integer, and is given by

We observe that the inequality (12) is equivalent to

Since , the class is a generalization of the class .

For some recent investigations on the class of close-to-convex functions, one can find them in [4–7] and the references cited therein. In the present paper, we aim at proving that the class is a subclass of meromorphic close-to-convex functions. Furthermore, some interesting results of the class are derived.

#### 2. Preliminary Results

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

Lemma 2. *Let , where . Then for , one has
*

*Proof. * Since , we have
We now let
Differentiating (17) logarithmically, we obtain
From (18) together with (16), we can get
Thus, if , we know that

Lemma 3 (see [8]). *Let . Then
*

Lemma 4 (see [9]). *Suppose that . Then
*

Lemma 5 (see [10, page 105]). *If the function
**
analytic and convex in and satisfies the condition
**
then
*

Lemma 6 (see [10]). *If the function is given by (3), then
*

Lemma 7 (see [11]). *Suppose that
**
Then
**
Each of these inequalities is sharp, with the extremal function given by
*

Lemma 8 (see [12]). *Let
**
be analytic in and let
**
be analytic and convex in . If , then
*

Lemma 9. *If , where is given by (3), then
*

*Proof. *By Lemma 8, we easily get the assertion of Lemma 9.

#### 3. Main Results

We first give the following result.

Theorem 10. *Let , Then
**
where is given by (13). *

*Proof. *From (13), we know that
Now, suppose that
Then, by Lemma 2 and (35), we get the assertion of Theorem 10 easily.

*Remark 11. *From Theorem 10 and Definition 1, we know that if , then is a meromorphic close-to-convex function. So the class is a subclass of meromorphic close-to-convex functions.

Now, we prove a sufficient condition for functions to belong to the class .

Theorem 12. *Let . If for , one has
**
where the coefficients () are given by (34), then . *

*Proof. *We set for given by (1) and defined by (13)
For , from inequality (37), we have
Thus, we have
that is, . This completes the proof of Theorem 12.

Next, we give the inclusion relationship for class .

Theorem 13. *Let . Then one has
*

*Proof. *Suppose that . By Definition 1, we have
Since , we get
Thus, by Lemma 3, we obtain
that is, . This means that . Hence the proof is completed.

In what follows, we derive the coefficient inequality for the class .

Theorem 14. *Suppose that
**
Then
*

*Proof. *Suppose that ; we know that
where is given by (34). If we set
it follows that
In view of Definition 1 and Lemma 9, we know that
By substituting the series expressions of functions , and into (49), we obtain
Since is univalent in , it is well known that
On the other hand, we find from (52) that
Combining (28), (51), and (54), we have
Thus, the assertion (47) of Theorem 14 follows directly from (55).

Finally, we give the distortion and growth theorems for the function class .

Theorem 15. *If , then
*

*Proof. *If , then there exists a function such that (12) holds. It follows from Theorem 10 that the function given by (34) is a meromorphic starlike function. Therefore, by Lemma 4, we have
Let be defined by (49); by Lemma 5, we know that
Thus, from (49), (58), and (59), we readily get (56). Upon integrating (56) from to , we get (57). The proof of Theorem 15 is thus completed.

#### Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant 11226088, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002, and the Science and Technology Program of Educational Department of Jiangxi Province under Grant GJJ12322 of China. The authors are grateful to the referees for their valuable comments and suggestions which essentially improved the quality of the paper.

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