Cofficient Estimates for a General Subclass of Bi-univalent Functions

Further, we shall denote by S the class of functions in A which are univalent in U(for details see [1, 3, 5]). Since univalent functions are one-to-one, they are invertible and inverse functions need not be defined on the entire unit disk U. The Koebe one-quarter theorem [5] ensures that the image of U under every univalent function f ∈ S contains a disk of radius 1 4 . So every function f ∈ S has an inverse f, which is defined by f(f(z)) = z (z ∈ U) and f(f(w)) = w (


Introduction
Let A denote the class of analytic functions in the unit disk U = {z ∈ C : |z| < 1}, that have the form: f (z) = z + ∞ n=2 a n z n . (1.1) Further, we shall denote by S the class of functions in A which are univalent in U(for details see [1,3,5]).
Since univalent functions are one-to-one, they are invertible and inverse functions need not be defined on the entire unit disk U. The Koebe one-quarter theorem [5] ensures that the image of U under every univalent function f ∈ S contains a disk of radius 1 4 . So every function f ∈ S has an inverse f −1 , which is defined by f −1 (f (z)) = z (z ∈ U) and f (f −1 (w)) = w |w| < r 0 (f ), r 0 (f ) ≥ 1 4 .
In fact, the inverse function f −1 is given by (1. 2) A function f ∈ A is said to be bi-univalent in U, if both f and f −1 are univalent in U (see [10]). Let Σ denote the class of bi-univalent functions in U given by (1.1). The class of bi-univalent functions was first introduced and studied by Lewin [6], where it was proved that |a 2 | ≤ 1.51.
Brannan and Taha [1](see also [2]), also investigated certain subclasses of bi-univalent functions and found non-sharp estimates on the first two Taylor-Maclurin coefficients |a 2 | and |a 3 |. For a brief history and interesting examples of functions in the class Σ, see [10].
Netanyahu [8], showed that max f ∈Σ The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients |a n | for n = 3, 4, ... is presumably still an open problem.
The object of the present paper is to introduce a new subclass of the function class Σ and find estimates on the coefficients |a 2 | and |a 3 | for functions in this new subclass of the functions class Σ employing the techniques used earlier by Srivastava et al. (see [9]).
, if the following conditions are satisfied: where g is the extension of f −1 to U.
. A function f (z) given by (1.1) is said to be in the S Σ (β, λ) (0 ≤ β < 1, 0 ≤ λ ≤ 1), if the following conditions are satisfied: and Re where g is the extension of f −1 to U. It is stated that in Theorem 3.1 in [7], the calculations done by Magesh for the bound |a 3 | are inaccurate. To remove this remarkable mistake, we've revised the calculations appropriately (see Theorem1.2). 2. Coefficient bounds for the function class S h,p Σ (A, B, C, λ) In this section, we introduce the subclass S h,p Σ (A, B, C, λ) (0 ≤ λ ≤ 1) and find the estimates on the coefficients |a 2 | and |a 3 | for functions in this subclass.
Also, let the continuous functions A, B, C : [0, 1] → R be so constrained that , if the following conditions are satisfied: and where g is the extension of f −1 to U.
it is easy to verify that the functions h(z) and p(z) satisfy the hypotheses of Definition 2.1. If In this case, the function f is said to be in the class S Σ (α, λ) introduced and studied by Magesh and Yamini [7].
If we get it is easy to verify that the functions h(z) and p(z) satisfy the hypotheses of Definition 2. and Re In this case, the function f is said to be in the class S Σ (β, λ) introduced and studied by Magesh and Yamini [7].
Proof. First of all, we write the argument inequalities in (2.1) and (2.2) in their equivalent forms as follows: and respectively, where functions h(z) and p(w) satisfy the conditions of Defintion 2.1. Furthermore, the functions h(z) and p(w) have the following Taylor-Maclaurin series expensions: .
(2.13) Therfore, we find from the equations (2.12) and (2.13) that respectively. So we get the desired estimate on the coefficient |a 2 | asserted. Next, in order to find the bound on the coefficient |a 3 |, we subtract (2.10) from (2.8). We thus get (2.14) Upon substituting the value of a 2 2 from (2.12) into (2.14), it follows that We thus find that If (3 + 3A − C) + (C − 2)(2A + C) = 0, then by substituting the value of a 2 2 from (2.13) into (2.14), it follows that . (2.16) Consequently, we have
By setting λ = 1 2 in Corollary 3.5, we conclude the following corollary.