Some Basic Properties of Certain New Subclass of Meromorphic Functions

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.