Coefficient problem for certain subclasses of bi-univalent functions defined by convolution

In this paper, we consider a general subclass H Σ (h, β) of bi-univalent functions. Bounds on the first two coefficients |a2| and |a3| for functions in H Σ (h, β) are given. The main results generalize and improve a recent one obtained by Srivastava [18].


Introduction
Let A denote the class of functions f which are analytic in the open unit disk U = {z : |z| < 1} with in the form (1) f (z) = z + ∞ n=2 a n z n .
Let S be the subclass of A consisting of the form (1) which are also univalent in U . For f (z) defined by (1) and Φ(z) defined by the Hadamard product (f * Φ)(z) of the functions f (z) and Φ(z) defined by a n Φ n z n For 0 ≤ β < 1 and λ ∈ C, we let Q λ (h, β) be the subclass of A consisting of functions f (z) of the form (1) and functions h(z) given by are also not members of Σ (see [18]).
In [16] the authors defined the classes of functions P m (β) : let P m (β), with m ≥ 2 and 0 ≤ β < 1, denote the class of univalent analytic functions P , normalized P (0) = 1, and satisfying For β = 0, we denote P m = P m (0), hence the class P m represents the class of functions p analytic in U , normalized with p(0) = 1, and having the representation where µ is a real valued function with bounded variation, which satisfies Clearly, P = P 2 is the well known class of Caratheodory functions, i.e. the normalized functions with positive real part in U .
, if the following conditions are satisfied: where the function h(z) is given by (4), a number λ ∈ C and (f * h) −1 (w) are defined by: We note that for λ = 1, m = 2 and h(z) = z 1−z , the class H λ Σ (h, β) reduce to the class H Σ (β) studied by Srivastava et al. [18].
The object of the present paper is to find for the first two coefficients |a 2 | and |a 3 | for functions in H λ Σ (h, β). The main results generalize and improve a recent one obtained by Srivastava [18].
In order to derive our main results, we require the following lemma. h n z n , z ∈ U , such that ϕ ∈ P m (β). Then |h n | ≤ m(1 − β), n ≥ 1.
Taking λ = 0 and λ = 1 in Theorem 2.1 we get following special cases, respectively.