Coeﬃcient Estimates for Certain Subclasses of Bi-Univalent Functions

Let Σ denote the class of bi-univalent functions in D = { z ∈ C : | z | < 1 } . In this paper, we consider two subclasses of Σ deﬁned in the open unit disk D which are denoted by S ∗ s, Σ ( φ ) and C s, Σ ( φ ). Besides, we ﬁnd up-per bounds for the second and third coeﬃcients for functions in these subclasses.


Introduction
Let A denote the class of functions f (z) normalized by the following Taylor-Maclaurin series: which are analytic in the open unit disk D = {z ∈ C : |z| < 1}. Further, let S denote the subclass of functions in A which are univalent in D. Some of the important and well-investigated subclasses of S include the class of starlike functions and the class of convex functions which are denoted by S * and C respectively. By definition, we have and C = f : f ∈ A and Re 1 + It readily follows from definitions (2) and (3) that The Koebe one-quarter theorem [4] states that the image of D under every function f (z) from S contains a disk of radius 1 4 . Thus every function f (z) ∈ S has an inverse f −1 (f (z)) defined by f −1 (f (z)) = z (z ∈ D) and In fact, the inverse function f −1 (w) is given by Let Σ denote the class of bi-univalent functions given by the Taylor-Maclaurin series expansion (1). Some examples of function in the class Σ are z 1−z , − log(1 − z) and 1 2 log 1+z 1−z . However, the familiar Koebe function is not a member of Σ. Other examples of function in S such as z − z 2 2 and 1 1−z 2 are also not members of Σ.
Lewin [5] investigated the class Σ and showed that |a 2 | < 1.51. Subsequently, Brannan and Clunie [1] conjectured that |a 2 | ≤ √ 2 for f ∈ Σ. Netanyahu [7], on the other hand, showed that max f ∈Σ |a 2 | = 4 3 . Brannan and Taha [2] introduced certain subclasses of Σ similar to the familiar subclasses of S consisting of strongly starlike, starlike and convex functions. They introduced bi-starlike functions and obtained estimates on the initial coefficients. The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients: is still an open problem.
If the functions f (z) and g(z) are analytic in D then f (z) is said to be subordinate to g(z) written as In [6], the authors introduced the class S * (φ) of Ma-Minda starlike functions and the class C(φ) of Ma-Minda convex functions, unifying previously studied classes related to starlike and convex functions. The class S * (φ) consists of all the functions f ∈ A satisfying the subordination zf (z) The function φ is analytic and univalent function with positive real part in D with φ(0) = 0, φ (0) > 0 and φ maps the unit disk D onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor's series expansion of such function is of the form where all coefficients are real and B 1 > 0.
In [10], Sakaguchi introduced the class S * s of starlike functions with respect to symmetric points in D, consisting of functions f ∈ A that satisfy the condition and in [3], Das and Singh introduced the class C s of convex functions with respect to symmetric points in D, consisting of functions f ∈ A that satisfy the condition Motivated by the earlier works of [10], [3] and [6] and considering functions f ∈ Σ, this paper introduce two subclasses of Σ and find estimates on the coefficients |a 2 | and |a 3 | for functions in these subclasses.

Preliminary Result and Definitions
In order to derive our main results, we need the following lemma.
Definition 2.1. A function f (z) ∈ Σ is said to be in class S * s,Σ (φ) if the following subordinations hold: and where g(w) = f −1 (w) is given by (5).

Main Results
For functions in the class S * s,Σ (φ), the following result is obtained. and Proof. Let f ∈ S * s,Σ (φ) and g = f −1 . Then there are analytic functions u, v : D → D, with u(0) = v(0) = 0, satisfying and Define the functions r 1 and r 2 by and v(z) = r 2 (z) − 1 r 2 (z) + 1 Then r 1 and r 2 are analytic in D with r 1 (0) = 1 = r 2 (0). Since u, v : D → D, the functions r 1 and r 2 have a positive real part in D and |b i | ≤ 2 and |c i | ≤ 2.
By subtracting (24) from (22), further computation using (21) and (25) lead to 8 and this yields the estimate given in (12). The proof of Theorem 3.1 is completed.
The result in Theorem 3.1 is similar to Theorem 2.3 in [8] if α = 0.
By using the similar approach as Theorem 3.1, we obtain the following result for functions f ∈ C s,Σ (φ).
Further computation using (30)-(34) lead to and this yields the estimate given in (27). The proof of Theorem 3.2 is completed.
The result in Theorem 3.2 is similar to Theorem 2.3 in [8] if α = 1.
For functions in the class S * s,Σ (φ), we obtained the result on Fekete-Szegö inequalities as follows.
Finally, we give the result on Fekete-Szegö inequalities for functions in the class C s,Σ (φ).