Certain subclasses of bi-univalent functions associated with the Hohlov operator

Lemma 1.3. (see [13]) If h ∈P, then |ck| ≦2 for each k, where P is the family of all functions h, analytic in U, for which ℜ{h(z)} > 0 (z ∈U), where h(z) = 1 + c1z + c2z2 + · · · (z ∈U). 2 Coefficient Bounds for the Function Class Sa,b;c Σ (α, λ) We begin by finding the estimates on the coefficients |a2| and |a3| for functions in the class Sa,b;c Σ (α, λ).

Theorem 2.1. Let the function f(z) given by (1) be in the following class: Sa,b;c Σ (α, λ) (0 < α ≦1; 0 ≦λ ≦1). Then |a2| ≦ 2α p [2α(λ2 −2λ) + (1 −α)(2 −λ)2]ϕ2 2 + 2α(3 −λ)ϕ3 (13) and |a3| ≦ 2α

Corollary 2.2. Let the function f(z) given by (1) be in the class Sa,b;c Σ (α) (0 < α ≦1). Then |a2| ≦α s 2 2(1 −α)ϕ2 2 + 3αϕ3 (29) and |a3| ≦2α 3ϕ3 . (30)

Corollary 2.3. Let the function f(z) given by (1) be in the class Hα Σ (0 < α ≦1). Then |a2| ≦α r 2 2 + α (31) and |a3| ≦2α 3 . (32)

Corollary 2.5. Let the function f(z) given by (1) be in the class Sa,b,c Σ (α, 1) (0 < α ≦1). Then |a2| ≦ 2α p (1 −3α)ϕ2 2 + 4αϕ3 and |a3| ≦α ϕ3 . 3 Coefficient Bounds for the Function Class Ma,b;c Σ (β, λ)

Theorem 3.1. Let the function f(z) given by (1) be in the following class: Ma,b;c Σ (β, λ) (0 ≦β < 1; 0 ≦λ ≦1). Then |a2| ≦ s 2(1 −β) (λ2 −2λ)ϕ2 2 + (3 −λ)ϕ3 (33) and |a3| ≦2(1 −β) (3 −λ)ϕ3 .

Corollary 3.2. Let the function f(z) given by (1) be in the class Ma,b,;c Σ (β) (0 ≦β < 1). Then |a2| ≦ s 2(1 −β) 3ϕ3 (47) and |a3| ≦2(1 −β) 3ϕ3 . (48)

Corollary 3.4. Let the function f(z) given by (1) be in the class Sa,b;c Σ (β) (0 ≦β < 1). Then |a2| ≦ s 2 −2β 2ϕ3 −ϕ2 2 and |a3| ≦1 −β ϕ3 .