International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 682162, 12 pages

doi:10.1155/2012/682162

Research Article

A Subclass of Meromorphic Close-to-Convex Functions of Janowski's Type

Young Jae Sim and Oh Sang Kwon

Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea

Received 9 April 2012; Accepted 19 June 2012

Academic Editor: Aloys Krieg

Copyright © 2012 Young Jae Sim and Oh Sang Kwon. 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 subclass of functions which are analytic in the punctured unit disk and meromorphically close-to-convex. We obtain some coefficients bounds and some argument and convolution properties belonging to this class.

1. Introduction

Let be the set of all analytic functions on the open unit disk , and let be the subclass of which contains the functions normalized by and . A function is said to be starlike of order if and only if

for some and for all . The class of starlike functions of order is denoted by .

If and are analytic in , we say that is subordinate to , written as follows:

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

such that

In particular, if the function is univalent in , the above subordination is equivalent to

The Hadamard product (or convolution) of two series

is defined by

For a convex function , it follows from Alexander’s Theorem that is a starlike function. In view of the identity , it is then clear that the classes of convex and starlike functions can be unified by considering functions satisfying is starlike for a fixed function .

Though the convolution of two univalent (or starlike) functions does not need be univalent, it is well-known that the classes of starlike, convex, and close-to-convex functions are closed under convolution with convex functions. These results were later extended to convolution with prestarlike functions. For , the class of prestarlike functions of order is defined by

while consists of satisfying .

By using the convex hull method [1, 2] and the method of differential subordination [3], Shanmugam [4] introduced and investigated convolution properties of various subclasses of analytic functions. Ali et al. [5] and Supramaniam et al. [6] investigated these properties for subclasses of multivalent starlike and convex functions. And Chandrashekar et al. [7] also investigated these properties for the functions with respect to symmetric points, conjugate, or symmetric conjugate points. More results using the convex hull method and the method of differential subordination can be found in [8, 9].

Let denote the class of all univalent meromorphic functions normalized by

which are analytic in the punctured unit disk

We denote by the subclass of consisting of formed by

which are meromorphic starlike of order in . In particular, we denote by , when . Also, a function of the form (1.9) is said to be meromorphic close-to-convex in if there is a in such that

We denote the set of functions close-to-convex in .

In a recent paper, Gao and Zhou [10] introduced an interesting subclass of the analytic function class and the univalent function , which contains the functions satisfying the following inequality:

for some . Here, is the class of starlike functions of order in . After that, many classes related to investigated and studied by some authors. Especially, Wang et al. [11, 12], Kowalczyk and Leś-Bomba [13], Xu et al. [14], and Seker [15] introduced the generalized class of the class . And they gave some properties of analytic functions in each classes.

In this paper, we define a class of meromorphic functions of Janowski’s type, related to the meromorphically close-to-convex functions as follows.

Definition 1.1. Let and be given by (1.9) and . Then , if there exists such that for ,

Theorem A. If , then .

Proof. At first, we know that

for , since . And let . Then

So we have

Hence .

Therefore, we know that is a subclass of , by Theorem A.

We will obtain some coefficient bounds and some argument and convolution properties belonging to this class.

2. On the Coefficient Estimates of Functions in

Now we give the coefficient estimates of functions in . We need the following Lemma to estimate of coefficient of functions in .

Lemma 2.1. Let and be given by (1.11), then one has

Proof. According to the Theorem A, we have , and if let

we know that , so is an odd meromorphic starlike function. If we let

it’s well-known [16, Volume 2, page 232] that

Substituting the series expressions of , in (2.2) and comparing the coefficients of two side of this equation, using (2.3) we can get an equality:

and the proof of Lemma 2.1 is completed.

Theorem 2.2. Let given by (1.9) and . If

then .

Proof. Let the functions and be given by (1.9) and (1.11), respectively. Furthermore, we let . Then

where is given by (2.5) and . Now we obtain

Thus, for , we have from (2.6)

Thus we have

which is equivalent to

which implies that .

Theorem 2.3. Let and is given by (1.9) and (1.11), respectively. Then for one has

where is given by (2.5).

Proof. Let . Then we have

where is an analytic function in , for and . Then,

Thus, putting

we obtain

With comparing the coefficients, we can write

Then, squaring the modulus of the both sides of the above equality and integrating along and using the fact the , we obtain

Letting on the both sides, we obtain

which implies the inequality (2.12).

Lemma 2.4 (see [2, Ruscheweyh and Sheil-Small]). Let and be convex in and suppose . Then .

Theorem 2.5. If , then there exists such that for all and with and ,

Proof. By definition, for , there exist and such that ,

Put , then

And (2.22) implies that

Since ,

hence

For and such that , , the function

is convex in . Applying Lemma 2.4 with this , we have

Given any function analytic in with , we have

so (2.27) reduces to

Exponentiating both sides of (2.29) leads to (2.20).

3. Argument and Convolution Properties of Functions in

In this section, we will solve some problems related to argument and convolution properties of functions in the class . To solve these problems, we need the following Lemmas.

Lemma 3.1 (see [17, Goluzin], [3, page 62]). Let be an analytic function in which is normalized and starlike, then

This inequality is sharp with extremal function .

Lemma 3.2 (see [1, Ruscheweyh]). Let , and . Then

for any analytic function , where denote the closed convex hull of .

Theorem 3.3. Let . Then

Proof. Let . Then there exists a function satisfying

where

And is contained in the disk

from which it follows that

Hence we can obtain the inequality:

Hence it suffices to find the upper bounds of . Since , we have for and . If we define , then and we can apply Lemma 3.1 to obtain

From the relation between and we obtain

which implies

Hence

for , and so the result is proved.

Theorem 3.4. Let satisfying the condition

and with . Then .

Proof. Let such that

and define the functions and by

Thus for any fixed ,

From the condition (3.13), we can easily obtain that

and this inequality implies that . Put . Since is an odd meromorphic starlike function, so is. Define a function by . Since is an odd meromorphic starlike function, and . Hence

Since and , Lemma 3.2 yields

where denote the closed convex hull of , and because

and is convex in , it follows that

in . Thus and the proof of Theorem 3.4 is completed.

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