Coefficient Estimates for New Subclasses of Meromorphic Bi-Univalent Functions

Abstract

We introduce and investigate two new subclasses and of meromorphic bi-univalent functions defined on . For functions belonging to these classes, estimates on the initial coefficients are obtained.

1. Introduction

Let denote the class of all functions of the form which are analytic in the open unit disk We also denote by the class of all functions in the normalized analytic function class which are univalent in .

Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk . In fact, the Koebe one-quarter theorem [1] ensures that the image of under every univalent function contains a disk of radius . Thus, every function has an inverse , which is defined by In fact, the inverse function is given by

A function is said to be bi-univalent in if both and are univalent in . Let denote the class of bi-univalent functions in given by (1). For a brief history and interesting examples of functions in the class , see [2] (see also [3, 4]). In fact, the aforecited work of Srivastava et al. [2] essentially revived the investigation of various subclasses of the bi-univalent function class in recent years; it was followed by such works as those by Murugusundaramoorthy et al. [5], Frasin and Aouf [6], Çağlar et al. [7], and others (see, e.g., [8–15]).

In this paper, the concept of bi-univalency is extended to the class of meromorphic functions defined on For this purpose, let denote the class of all meromorphic univalent functions of the form defined on the domain . Since is univalent, it has an inverse that satisfies Furthermore, the inverse function has a series expansion of the form where . Analogous to the bi-univalent analytic functions, a function is said to be meromorphically bi-univalent if both and are meromorphically univalent in . We denote by the class of all meromorphic bi-univalent functions in given by (6). A simple calculation shows that

The coefficient problem was investigated for various interesting subclasses of the meromorphic univalent functions (see, e.g., [16–18]).

In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficients and of functions in the newly introduced subclasses are obtained.

Definition 1. A function given by (6) is said to be in the class if the following conditions are satisfied: where the function is given by (9).

Definition 2. A function given by (6) is said to be in the class if the following conditions are satisfied: where the function is given by (9).

Remark 3. We note that, for , the classes and reduce to the classes respectively, introduced and studied by Halim et al. [17].

The object of the present paper is to extend the concept of bi-univalent to the class of meromorphic functions defined on and find estimates on the coefficients and for functions in the above-defined classes and of the function class .

Firstly, in order to derive our main results, we need the following lemma.

Lemma 4 (see [19]). If , then for each , where is the family of all functions analytic in for which for .

2. Coefficient Estimates

We begin this section by finding the estimates on the coefficients and for functions in the class .

Theorem 5. Let the function given by the series expansion (6) be in the function class Then

Proof. It follows from (10) that respectively, where and are functions with positive real part in and have the forms respectively. Now, upon equating the coefficients in (17), we get From (20) and (22), we find that Also, from (21) and (23) we obtain Since and in , the functions and hence the coefficients and for each satisfy the inequality in Lemma 4. Applications of triangle inequality followed by Lemma 4 in (25) and (26) give us the required estimates on as asserted in (15).
Next, in order to find the bound on the coefficient , we subtract (23) from (21). We thus get Hence On the other hand, using (21) and (23) yields By using (25) we have from the above equality From Lemma 4, we obtain Also, by using (26) we have, from equality (29), Comparing (28), (31), and (32) we get the desired estimate on the coefficient as asserted in (16).

For , we have the following corollary of Theorem 5.

Corollary 6. Let the function given by the series expansion (6) be in the function class Then

Remark 7. Corollary 6 is an improvement of the following estimates which were given by Halim et al. [17].

Corollary 8 (see [17]). Let the function given by the series expansion (6) be in the function class Then

Remark 9. Corollary 6 is also an improvement of the estimates which were given by Panigrahi [20, Corollary ].

Next we estimate the coefficients and for functions in the class .

Theorem 10. Let the function given by the series expansion (6) be in the function class Then

Proof. It follows from (11) that respectively, where and are functions with positive real part in and have the forms (18) and (19), respectively. Now, upon equating the coefficients in (40), we get From (41) and (43), we obtain Also, from (42) and (44), we obtain
Since and in , the functions and hence the coefficients and for each satisfy the inequality in Lemma 4. Therefore, we find from (46) and (47) that respectively. So we get the desired estimate on the coefficient as asserted in (38).
Next, in order to find the bound on the coefficient , we subtract (44) from (42). We thus get Hence On the other hand, using (42) and (44) yields or equivalently Upon substituting the value of from (46) and (47) into (52), respectively, it follows that Comparing (50) and (53), we get the desired estimate on the coefficient as asserted in (39).

For , we have the following corollary of Theorem 10.

Corollary 11. Let the function given by the series expansion (6) be in the function class Then

Remark 12. Corollary 11 is an improvement of the following estimates which were given by Halim et al. [17].

Corollary 13 (see [17]). Let the function given by the series expansion (6) be in the function class Then

Remark 14. Corollary 11 is also an improvement of the estimates which were given by Panigrahi [20, Corollary ].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.