Coefficient Estimates for New Subclasses of Meromorphic Bi-Univalent Functions

We introduce and investigate two new subclasses ℳ σ(α, λ) and ℳ σ(β, λ) of meromorphic bi-univalent functions defined on Δ = {z : z ∈ ℂ, 1 < |z | <∞}. For functions belonging to these classes, estimates on the initial coefficients are obtained.


Introduction
Let A denote the class of all functions of the form which are analytic in the open unit disk U = { : ∈ C, | | < 1} .
We also denote by S the class of all functions in the normalized analytic function class A which are univalent in U.
Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk U. In fact, the Koebe one-quarter theorem [1] ensures that the image of U under every univalent function ∈ S contains a disk of radius 1/4. Thus, every function ∈ A has an inverse −1 , which is defined by −1 ( ( )) = ( ∈ U) , ( −1 ( )) = (| | < 0 ( ) ; 0 ( ) ≥ 1 4 ) . ( In fact, the inverse function −1 is given by A function ∈ A is said to be bi-univalent in U if both and −1 are univalent in U. Let Σ denote the class of biunivalent functions in U given by (1). For a brief history and interesting examples of functions in the class Σ, 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 Σ in recent years; it was followed by such works as those by Murugusundaramoorthy et al. [5], Frasin and Aouf [6], Ç aglar et al. [7], and others (see, e.g., [8][9][10][11][12][13][14][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 Δ. Since ∈ is univalent, it has an inverse −1 that satisfies 2 International Scholarly Research Notices Furthermore, the inverse function −1 has a series expansion of the form where < | | < ∞. Analogous to the bi-univalent analytic functions, a function ∈ is said to be meromorphically biunivalent if both and −1 are meromorphically univalent in Δ. We denote by M the class of all meromorphic bi-univalent functions in Δ 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][17][18]).
In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficients 0 and 1 of functions in the newly introduced subclasses are obtained. (6) is said to be in the class M ( , ) if the following conditions are satisfied:

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

Definition 2. A function given by
where the function ℎ = −1 is given by (9).
The object of the present paper is to extend the concept of bi-univalent to the class of meromorphic functions defined on Δ and find estimates on the coefficients | 0 | and | 1 | for functions in the above-defined classes M ( , ) and M ( , ) of the function class M .
Firstly, in order to derive our main results, we need the following lemma.
Lemma 4 (see [19]). If ∈ P, then | | ≤ 2 for each , where P is the family of all functions analytic in U for which for ∈ U.

Coefficient Estimates
We begin this section by finding the estimates on the coefficients | 0 | and | 1 | for functions in the class M ( , ). Then Proof. It follows from (10) that respectively, where ( ) and ( ) are functions with positive real part in Δ and have the forms respectively. Now, upon equating the coefficients in (17), we get International From (20) and (22), we find that Also, from (21) and (23) we obtain Since R( ( )) > 0 and R( ( )) > 0 in Δ, the functions (1/ ), (1/ ) ∈ P 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 | 0 | as asserted in (15).
Next, in order to find the bound on the coefficient | 1 |, we subtract (23) from (21). We thus get 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 | 1 | as asserted in (16).
\ For = 0, 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 0 ≤ 2 , Remark 9. Corollary 6 is also an improvement of the estimates which were given by Panigrahi [20, Corollary 2.3].
Next we estimate the coefficients | 0 | and | 1 | for functions in the class M ( , ). (37) Proof. It follows from (11) that respectively, where ( ) and ( ) are functions with positive real part in Δ and have the forms (18) and (19), respectively. Now, upon equating the coefficients in (40), we get 4 International Scholarly Research Notices From (41) and (43), we obtain Also, from (42) and (44), we obtain Since R( ( )) > 0 and R( ( )) > 0 in Δ, the functions (1/ ), (1/ ) ∈ P 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 | 0 | as asserted in (38). Next, in order to find the bound on the coefficient | 1 |, we subtract (44) from (42). We thus get On the other hand, using (42) Upon substituting the value of 2 0 from (46) and (47) into (52), respectively, it follows that Comparing (50) and (53), we get the desired estimate on the coefficient | 1 | as asserted in (39).
For = 0, we have the following corollary of Theorem 10.
Remark 12. Corollary 11 is an improvement of the following estimates which were given by Halim et al. [17].