Journal of Applied Mathematics

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

doi:10.1155/2012/494917

Research Article

A New Class of Meromorphic Functions Associated with Spirallike Functions

Lei Shi, Zhi-Gang Wang, and Jing-Ping Yi

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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