Estimating coefficients for certain subclasses of meromorphic and bi-univalent functions

In the present paper, we introduce two interesting subclasses of meromorphic and bi-univalent functions defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta=\{z:z\in\mathbb{C}, 1<|z|<\infty\}$\end{document}Δ={z:z∈C,1<|z|<∞}. For functions belonging to these subclasses, estimates on the initial coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|b_{0}|$\end{document}|b0| and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|b_{1}|$\end{document}|b1| are obtained. Some other closely related results are also represented. The coefficient bounds presented here are new in their own kind. We hope that this paper will generate further interest in applying our approach to other related problems.


Introduction and lemmas
Let A be the class of functions of the form f (z) = z + ∞ k=2 a k z k , (1.1) which are analytic in the open unit disk U = {z ∈ C : |z| < 1}. We also denote by S the subclass of the normalized analytic function class A consisting of all functions in which are also 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 [3] ensures that the image of U under every univalent function f ∈ S contains a disk of radius 1/4. Thus every function f ∈ S has an inverse f -1 , which is defined by , and f f -1 (w) = w |w| < r 0 (f ); r 0 (f ) ≥ 1 4 .
A function f ∈ A is said to be bi-univalent in the open unit disk U if both the function and its inverse are univalent in U. Let σ denote the class of analytic and bi-univalent functions in U given by the Taylor-Maclaurin series expansion as in (1.1). For a brief history and interesting examples of functions in the class σ , see [19]. In fact, the aforecited work of Srivastava et al. [19] essentially revived the investigation of various subclasses of the biunivalent function class σ in recent years; it was followed by works of, e.g., Frasin and Aouf [6], Srivastava et al. [18,20], Xu et al. [21,22] and others (see, for example, [1,2,4,7,11,14]).
In this paper, the concept of bi-univalency is extended to the class of meromorphic functions defined on = {z : z ∈ C, 1 < |z| < ∞}. The class of functions that are meromorphic and univalent in is denoted by , and every univalent function g has an inverse g -1 satisfying the series expansion where 0 < M < |w| < ∞. Analogous to the bi-univalent analytic functions, a function g ∈ is said to be meromorphic and bi-univalent if both g and g -1 are meromorphic and univalent in given by (1.2). The class of all meromorphic and bi-univalent functions is denoted by σ . A simple calculation shows that Estimates on the coefficients of the inverses of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer [15] showed that if g, defined by (1.2), is in with b 0 = 0, then |b 2 | ≤ 2/3. In 1971, Duren [5] obtained the inequality |b n | ≤ 2/(n + 1) for g ∈ with b k = 0, 1 ≤ k < n/2. For g -1 being the inverse of g, Springer [17] showed that and conjectured that In 1977, Kubota [9] proved that the Springer conjecture is true for n = 3, 4, 5, and subsequently Schober [16] obtained sharp bounds for the coefficients B 2n-1 (1 ≤ n ≤ 7). Recently, Kapoor and Mishra [8] found coefficient estimates for inverses of meromorphic starlike functions of positive order α in .
In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficients |b 0 | and |b 1 | of functions in the newly introduced subclasses are obtained. These coefficient results are obtained by associating with the functions having positive real part. An analytic function p of the form p(z) = 1 + c 1 z + c 2 z 2 + · · · is called a function with positive real part in U if p(z) > 0 for all z ∈ U. The class of all functions with positive real part is denoted by P. We need the following lemmas [13] to prove our main results. Lemma 1.1 If φ(z) ∈ P, the class of functions analytic in U with positive real part, is given by then |c n | ≤ 2 for each n ∈ N.
In 1972, the following univalence criterion was proved by Ozaki and Nunokawa [12].
then f is univalent in U and hence f ∈ S.
Also, let T (μ) denote the class of functions f ∈ A such that where μ is a real number with 0 < μ ≤ 1 and T (1) = T . It is clear that T (μ) ⊂ T ⊂ S.

Main results
Definition 2.1 A function g ∈ σ given by (1.2) is said to be in the class T α σ if the following conditions are satisfied: where the function h is an extension of g -1 to defined by (1.4).

Definition 2.2
A function g ∈ σ given by (1.2) is said to be in the class T σ (μ) if the following conditions are satisfied: [g(z)] 2 > 1μ (z ∈ ; 0 < μ ≤ 1) and where the function h is an extension of g -1 to defined by (1.4).

Theorem 2.1 Let the function g, given by the series expansion (1.2), be in the function class
Proof It follows from (2.1) that respectively, where s(z) and t(z) are functions with positive real part in and have the forms and respectively. Now, upon equating the coefficients in (2.5), we get -2b 0 = αs 1 , (2.8) 2b 0 = αt 1 , (2.10) and, from (2.8) and (2.10), we find that Also from (2.9) and (2.11) we obtain (2.14) Since R(s(z)) > 0 and R(t(z)) > 0 in , the functions s(1/z), t(1/z) ∈ P and hence the coefficients s k and t k for each k satisfy the inequality in Lemma 1.1. Applying the triangle inequality, and then Lemma 1.1, in (2.13) and (2.14) gives us the desired estimates on |b 0 |, as asserted in (2.3).
Next, in order to find the bound on the coefficient |b 1 |, we subtract (2.11) from (2.9), and we thus get On the other hand, using (2.9) and (2.11) yields (2.17) By using (2.13), we have from the above equality   Since R(s(z)) > 0 and R(t(z)) > 0 in , the functions s(1/z), t(1/z) ∈ P and hence the coefficients s k and t k for each k satisfy the inequality in Lemma 1.1. Therefore we find from (2.30) and (2.31) that |b 0 | ≤ μ and |b 0 | ≤ 2μ 3 , (2.32) respectively. So we get the desired estimate on the coefficient |b 0 |, as asserted in (2.21). Next, in order to find the bound on the coefficient |b 1 |, we subtract (2.28) from (2.26), and obtain