Coefficient Bounds for Certain Subclasses of Bi-univalent Functions

under the Creative Commons Attribution License, which permits unrestricted use, distribution , and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we introduce and investigate two new subclasses of the function class Σ of bi-univalent functions. Also, we find estimates of |a 2 | and |a 3 |. Some related consequences of the results are also pointed out.


Introduction
Let A denote the class of functions of the form It is well known that every function f ∈ S has an inverse f −1 , defined by 1−z and so on.However, the familiar Koebe function is not a member of Σ.Other common examples of functions in S such as z − z 2 2 and z 1−z 2 are also not members of Σ (see [5,12]).
In 1967, Lewin [7] investigated the bi-univalent function class Σ and showed that |a 2 | < 1.51.Subsequently, Brannan and Clunie [2] conjectured that |a 2 | ≤ √ 2. Netanyahu [10], on the other hand, showed that max A function f ∈ A is in the class S α Σ of strongly bi-starlike of order α (0 < α ≤ 1), if each of the following condition is satisfied: , and arg wg (w) where the function g is given by The classes S * Σ (α) and K Σ (α) of bi-starlike functions of order α and biconvex functions of order α, corresponding to the function classes S * (α) and K(α) defined by (1.1.2) and (1.1.3),were also introduced analogously.For each of the function classes S * Σ (α) and K Σ (α), Brannan and Taha [4] found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 | (for details see [4,14]).Following Brannan and Taha [4], Srivastava et al. [12] introduced certain subclass H α Σ , 0 < α ≤ 1 of the bi-univalent functions class Σ, a function f (z) given by (1.1.1)is said to be in the class H α Σ , 0 < α ≤ 1, if the following conditions are satified: where the function g is given Then later many researchers (see [1,6,15,16]) studied extensively the same class H α Σ , by different techniques and found the non-sharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 |.It is interest to note that the estimates were found are improved but not sharp.Further, Frasin and Aouf [5] extended the class H α Σ , and obtained the non-sharp bounds (see also [9,13]).
Motivated by the aforementioned works, we introduce the following subclasses of the function class Σ. Definition 1.1.A function f (z) given by (1.1.1)is said to be in the class S Σ (α, λ) if the following conditions are satisfied: where the function g is given by 1.1.6.
Definition 1.2.A function f (z) given by (1.1.1)is said to be in the class M Σ (β, λ) if the following conditions are satisfied: where the function g is given by (1.1.6).
The object of the present paper is to find estimates on the coefficients |a 2 | and |a 3 | for functions in the above-defined subclasses S Σ (α, λ) and M Σ (α, λ) of the function class Σ.
In order to derive our main results, we shall need the following lemma.

Lemma 1.3. ([11]
) If h ∈ P, then |c k | 2 for each k, where P is the family of all functions h, analytic in U, for which Then and where p(z) and q(w) in P and have the following forms: respectively.Now, equating the coefficients in (2.2.3) and (2.2.4), we get (2.2.8) (2.2.10)From (2.2.7) and (2.2.9), we get ). (2.2.12)From (2.2.8), (2.2.10) and (2.2.12), we obtain Applying Lemma 1.3 for the coefficients p 2 and q 2 , we immediately have This gives the bound on |a 2 | as asserted in (2.2.1).Next, in order to find the bound on |a 3 |, by subtracting (2.2.10) from (2.2.8), we get It follows from (2.2.11), (2.2.12) and (2.2.13) that Applying Lemma 1.3 once again, we readily get This completes the proof of Theorem 2.1.
In the following section we find the estimates on the coefficients |a 2 | and |a 3 | for functions in the class M Σ (β, λ). and Proof.It follows from (1.1.9)and (1.1.10)that there exists p, q ∈ P such that and where p(z) and q(w) have the forms (2.2.5) and (2.2.6), respectively.Equating coefficients in (3.3.3) and (3.3.4),we get From (3.3.5) and (3.3.7),we get .
Applying Lemma 1.3 for the coefficients p 2 and q 2 , we immediately have This gives the bound on |a 2 | as asserted in (3.Taylor-Maclaurin coefficients obtained in [8].Also, for the choice of λ = 1 2 , the results stated in Theorem 2.1 and Theorem 3.1 would improve bounds stated in [12].

1 . 1 )
which are analytic in the open unit disc U = {z : z ∈ C and |z| < 1}.Further, by S we shall denote the class of all functions in A which are univalent in U.Some of the important and well-investigated subclasses of the univalent function class S include (for example) the class S * (α) of starlike functions of order α in U and the class K(α) of convex functions of order α in U.By definition, we haveS * (α) := f : f ∈ S and zf (z) f (z) > α; z ∈ U; 0 ≤ α < 1 (1.1.2) and K(α) := f : f ∈ S and 1 + zf (z) f (z) > α; z ∈ U; 0 ≤ α < 1 .(1.1.3)It readily follows from the definitions (1.1.2) and (1.1.3)that f ∈ K(α) ⇐⇒ zf ∈ S * (α).