Journal of Function Spaces
Volume 2015 (2015), Article ID 196060, 6 pages
http://dx.doi.org/10.1155/2015/196060

Research Article

Some Basic Properties of Certain New Subclass of Meromorphic Functions

Lei Shi¹ and Zhi-Gang Wang²

¹School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China
²School of Mathematics and Computing Science, Hunan First Normal University, Changsha, Hunan 410205, China

Received 13 December 2014; Accepted 11 February 2015

Academic Editor: Giuseppe Marino

Copyright © 2015 Lei Shi and Zhi-Gang Wang. 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 and investigate a new subclass of meromorphic functions. Some interesting properties such as inclusion relationship, coefficient estimates, neighborhoods, and partial sums are proved. Connections of the results with known results are also considered.

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 of the form

which are analytic and satisfy the condition

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

For , Wang et al. [1] (see also Nehari and Netanyahu [2]) introduced and studied a new subclass of consisting of functions satisfying

We note that meromorphic starlike functions and related topics attract many authors’ attentions; see (for example) the earlier works [3–8] and the references cited therein.

Let

be analytic in . Assuming that and , we say that a function if it satisfies the condition

The function class was introduced and studied recently by Ravichandran et al. [9], Liu et al. [10], Singh and Gupta [11], and Wang et al. [12].

In [13], Wang et al. introduced a subclass of meromorphic function which satisfies the condition

It was proved that the class is a subclass of the of meromorphically starlike functions of order .

Motivated essentially by the above works, we introduce and investigate a new subclass of of meromorphic functions.

Definition 1. Suppose that . Let denote a subclass of consisting of functions satisfying the condition that

We note that, for , the class reduces to .

In the present paper, we aim at proving some interesting properties such as inclusion relationship, coefficient estimates, neighborhoods, and partial sums for functions in the class .

The following lemmas will be required in our investigation.

Lemma 2 (see [14]). Let be a set in the complex plane and suppose that is mapping from to which satisfies for and for all real , such that . If the function is analytic in and for all , then .

Lemma 3. Let , , and . Suppose also that the sequence is defined by

Then

Proof. From (11), we have

Combining (13), we find that

Thus, for , we deduce from (14) that

This completes the proof of Lemma 3.

Lemma 4. Let

Suppose also that is given by (1) and

where (and throughout this paper unless otherwise mentioned) the parameter is defined as

Then .

The proof of Lemma 4 is similar to that of Theorem 1 in Wang et al. [1] and so is omitted.

2. Main Results

We begin by proving the following result which shows that is a subclass of .

Theorem 5. Suppose that and . Then

Proof. Define

Then is analytic in . It follows from (20) that

Combining (20) and (21), we obtain that

where

For all real and satisfying , we have

If we set

then for all real such that . Moreover, from definition (10), we know that . Using Lemma 2, we conclude that for all , which implies that . This completes the proof of Theorem 5.

Now we consider the coefficient estimates for functions belonging to the class .

Theorem 6. Suppose that

If , then

Proof. Suppose that . Then there exists such that

It follows from (28) that

Combining (1) and (29), we have

Evaluating the coefficient of in both sides of (30) yields

By observing the fact that for , we find from (31) and (32) that

Now we define the sequence as follows:

In order to prove that

we use the principle of mathematical induction by noting that

Therefore, we assume that

Combining (32) and (33), we get

Hence, by the principle of mathematical induction, we have

as desired. By means of Lemma 2 and (33), we know that (12) holds. Combining (39) and (12), we readily get the coefficient estimates asserted by Theorem 6.

Using Lemma 4, we introduce the -neighborhood of a function of the form (1) by means of the following definition:

By making use of definition (40), we obtain the following result.

Theorem 7. If satisfies the condition

then

Proof. It is easily seen from (10) that a function if and only if

which is equivalent to

where

It follows from (45) that

Furthermore, under the hypotheses of Theorem 7, (44) yields the following inequality:

Suppose that

It follows from (40) that

Combining (47) and (49), we have

which implies that

Thus, we have

This completes the proof of Theorem 7.

Finally, we derive the partial sums of functions in the class .

Theorem 8. Let be given by (1) and define the partial sums of by

Suppose also that

Then

(1); (2)

Each of the bounds in (55) and (56) is the best possible for each .

Proof. (1) It is easy to see that the result follows directly from Lemma 4.
(2) Note that

Thus, we have

By setting

we find from (58) and (59) that

which implies inequality (55).
If we put

then

which shows that the bound in (55) is the best possible for each .
Now, we set

In view of (58) and (63), we conclude that

which leads to inequality (56) asserted in Theorem 8. The bound in (56) is sharp with the extremal function given by (61). We thus complete the proof of Theorem 8.

In what follows, we turn to quotients involving derivatives. The proof of Theorem 9 is similar to that of Theorem 8 and so the details may be omitted.

Theorem 9. Let be given by (1) and define the partial sums of by (53). If the condition (54) holds, then

The bounds in (65) are sharp with the extremal function given by (61).

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

The authors jointly worked on the results and they read and approved the final paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008 and 11426035 and the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146.

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