Coefficients Estimates of the Class of Biunivalent Functions

Let A be the class of functions f which are analytic in the open unit disk D = {z ∈ C : |z| < 1} and normalized by the conditions f(0) = 0 and f󸀠(0) = 1. The Koebe one-quarter theorem [1] ensures that the image ofD under every univalent functionf ∈ A contains the disk with the center in the origin and the radius of 1/4. Thus, every univalent function f ∈ A has an inversef : f(D) → D, satisfyingf(f(z)) = z (z ∈ D) and

A function  ∈ A is said to be biunivalent in D if both  and  −1 are univalent in D, supposing that D ⊆ (D), and we denote the class of biunivalent functions by .
In [2], the author defined the following classes P  () as follows.
Lewin [3] investigated the class  of biunivalent functions and obtained the bound for the second coefficient.Brannan and Taha [4] considered certain subclasses of biunivalent functions, similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions.They introduced the bistarlike functions and the biconvex functions and obtained estimates on the initial coefficients.Recently, Ali et al. [5], Srivastava et al. [6], Frasin and Aouf [7], Goyal and Goswami [8], and many others have introduced and investigated subclasses of biunivalent functions and obtained bounds for the initial coefficients.In the papers of Jahangiri et al. [9] and Hamidi et al. [10], the authors use Faber polynomial to find upper bounds for   .In this paper, we will try to find upper bound of |  | for the class BR   (; ) which is defined below.
Definition 1.A function  ∈  is said to be in the class BR   (; ) if the following conditions are satisfied: where  =  −1 .

Main Results
In order to prove our main result for the functions class  ∈ BR   (; ), we first use the following lemma.

Lemma 3. Let the function Φ given by
Proof.Proof of this lemma is straightforward, if we write This gives Using known result [15] for class P  , we have our result.13) If  is of form (3), then Similarly, for  =  −1 given by ( 5), we have Consequently, for (  ())  , we have Now from Definition 1, there exist two functions () and () that belong to P(; ) such that Now comparing the coefficients of ( 15) and ( 18), the following is given: Similarly, from ( 17) and ( 19), If   = 0 for 2 ≤  ≤  − 1, then we can combine (20) and (21) and using relation (8), it yields Now taking absolute value in both sides of above equations and using Lemma (,  ∈ N, 0 ≤  < 1) . (24) Proof.Since  ∈ BR   (; ), from (3) and (4) we have where () ∈ P  () and () ∈ P  ().It is easy to see that the functions  and  have the following Taylor expansions: Now, equating the coefficients in (25), we get