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Journal of Applied Mathematics

Volume 2012 (2012), Article ID 494917, 12 pages

http://dx.doi.org/10.1155/2012/494917

## A New Class of Meromorphic Functions Associated with Spirallike Functions

School of Mathematics and Statistics, Anyang Normal University, Henan, Anyang 455002, China

Received 22 September 2012; Revised 26 October 2012; Accepted 31 October 2012

Academic Editor: Alberto Cabada

Copyright © 2012 Lei Shi 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 new class of meromorphic functions associated with spirallike functions. Such results as subordination property, integral representation, convolution property, and coefficient inequalities are proved.

#### 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 in and satisfy the condition

Let , where is given by (1.1) and is defined by then the Hadamard product (or convolution) is defined by

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:

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

For the real number , we know that If the complex number satisfies the condition it can be easily verified that

We now introduce and investigate the following class of meromorphic functions.

*Definition 1.1. *A function is said to be in the class if it satisfies the inequality

*Remark 1.2. *For , the class is the familiar class of meromorphic starlike functions of order .

*Remark 1.3. *If , then the condition (1.16) is equivalent to
which implies that belongs to the class of meromorphic spirallike functions. Thus, the class of meromorphic spirallike functions is a special case of the class .

For some recent investigations on spirallike functions and related functions, see, for example, the earlier works [1–9] and the references cited in each of these earlier investigations.

*Remark 1.4. *The function
belongs to the class .

It is clear that
Then, for the function given by (1.18), we know that
which implies that .

In this paper, we aim at deriving the subordination property, integral representation, convolution property, and coefficient inequalities of the function class .

#### 2. Preliminary Results

In order to derive our main results, we need the following lemmas.

Lemma 2.1. *Let be a complex number. Suppose also that the sequence is defined by
**
Then
*

*Proof. *From (2.1), we know that
By virtue of (2.3), we find that
Thus, for , we deduce from (2.4) that
By virtue of (2.1) and (2.5), we get the desired assertion (2.2) of Lemma 2.1.

Lemma 2.2 (Jack's Lemma [10]). *Let be a nonconstant regular function in . If attains its maximum value on the circle at , then
**
for some real number .*

#### 3. Main Results

We begin by deriving the following subordination property of functions belonging to the class .

Theorem 3.1. *A function if and only if
*

*Proof. *Suppose that
We easily know that , which implies that
where is analytic in with and .

It follows from (3.3) that
which is equivalent to the subordination relationship (3.1).

On the other hand, the above deductive process can be converse. The proof of Theorem 3.1 is thus completed.

Theorem 3.2. *Let . Then
**
where is analytic in with and .*

*Proof. *For , by Theorem 3.1, we know that (3.1) holds true. It follows that
where is analytic in with and .

We now find from (3.6) that
which, upon integration, yields
The assertion (3.5) of Theorem 3.2 can be easily derived from (3.8).

Theorem 3.3. *Let . Then
*

*Proof. *Assume that . By Theorem 3.1, we know that (3.1) holds, which implies that
It is easy to see that the condition (3.10) can be written as follows:
We note that
Thus, by substituting (3.12) into (3.11), we get the desired assertion (3.9) of Theorem 3.3.

Theorem 3.4. *Let . If , then
**
The inequality (3.13) is sharp for the function given by
*

*Proof. *Suppose that
We easily know that .

If we put
it is known that
From (3.15), we have
We now set
It follows from (3.18) that
Combining (1.1), (3.16), and (3.20), we obtain
In view of (3.21), we get
From (3.17) and (3.22), we obtain
Moreover, we deduce from (3.17) and (3.23) that

Next, we define the sequence as follows:

In order to prove that
we make use of the principle of mathematical induction. By noting that
Therefore, assuming that
Combining (3.25) and (3.26), we get
Hence, by the principle of mathematical induction, we have
as desired.

By means of Lemma 2.1 and (3.26), we know that
Combining (3.31) and (3.32), we readily get the coefficient estimates asserted by Theorem 3.4.

For the sharpness, we consider the function given by (3.14). A simple calculation shows that
Thus, the function belongs to the class . Since , we have
Then becomes
This completes the proof of Theorem 3.4.

Theorem 3.5. *If satisfies the inequality
**
then . *

*Proof. *To prove , it suffices to show that
which is equivalent to
From (3.36), we know that

Now, by the maximum modulus principle, we deduce from (1.1) and (3.39) that
which implies that the assertion of Theorem 3.5 holds.

Theorem 3.6. *If satisfies the condition
**
then .*

*Proof. *Define the function by
Then we see that is analytic in with .

It follows from (3.42) that
By differentiating both sides of (3.43) logarithmically, we obtain
From (3.41) and (3.44), we find that
Next, we claim that . Indeed, if not, there exists a point such that
By Lemma 2.2, we have
Moreover, for , we find from (3.44) and (3.47) that
But (3.48) contradicts to (3.45). Therefore, we conclude that , that is
which shows that .

#### Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grants 11101053 and 11226088, the Key Project of Chinese Ministry of Education under Grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under Grant 10B002, the Open Fund Project of the Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002 of the People’s Republic of China. The authors would like to thank the referees for their careful reading and valuable suggestions which essentially improved the quality of this paper.

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