Coefficient bounds for new subclasses of bi-univalent functions

Lemma 1.1. [7] If p ∈P, then |ck| ≤2 for each k, where P is the family of all functions p analytic in U for which ℜp(z) > 0, p(z) = 1 + c1z + c2z2 + ... for z ∈U. 2. Coefficient bounds for the function class N µ Σ (α, λ) Definition 2.1. A function f(z) given by (1.1) is said to be in the class N µ Σ (α, λ) if the following conditions are satisfied: (2.1) f ∈Σ and arg

Theorem 2.1. Let f(z) given by (1.1) be in the class N µ Σ (α, λ) , 0 < α ≤1, λ ≥1 and µ ≥0. Then (2.4) |a2| ≤ 2α q (λ + µ)2 + α µ + 2λ −λ2
and |a3| ≤ 4α2 (λ + µ)2 + 2α 2λ + µ.

Corollary 2.2. [4] Let f(z) given by (1.1) be in the class N 1 Σ (α, λ) , 0 < α ≤1 and λ ≥1. Then |a2| ≤ 2α q (λ + 1)2 + α 1 + 2λ −λ2
and |a3| ≤ 4α2 (λ + 1)2 + 2α 2λ + 1. If we choose λ = µ = 1 in Theorem 2.1, we get the following corollary.

Corollary 2.3. [8] Let f(z) given by (1.1) be in the class N 1 Σ (α, 1) , 0 < α ≤1. Then |a2| ≤α r 2 α + 2 and |a3| ≤α (3α + 2) 3 . If we choose λ = µ + 1 = 1 in Theorem 2.1, we obtain well-known the class N 0 Σ (α, 1) = S∗ Σ[α] of strongly bi-starlike functions of order α and get the following corollary.

Corollary 2.4. Let f(z) given by (1.1) be in the class S∗ Σ[α], 0 < α ≤1. Then |a2| ≤ 2α √1 + α and |a3| ≤α(4α + 1) . 3. Coefficient bounds for the function class N µ Σ (β, λ) Definition 3.1. A function f(z) given by (1.1) is said to be in the class N µ Σ (β, λ) if the following conditions are satisfied: (3.1) f ∈Σ and ℜ

Theorem 3.1. Let f(z) given by (1.1) be in the class N µ Σ (β, λ) , 0 ≤β < 1, λ ≥1 and µ ≥0. Then (3.3) |a2| ≤min (s 4 (1 −β) (µ + 1) (2λ + µ), 2 (1 −β) λ + µ ) and (3.4) |a3| ≤  

Theorem 2.1, by suitably comparing coefficients in (3.5) and (3.6), we get (λ + µ) a2 = (1 −β) p1, (3.7) (2λ + µ) a3 + (µ −1)  λ + µ 2  a2 2 = (1 −β) p2, (3.8)

Corollary 3.2. Let f(z) given by (1.1) be in the class N 1 Σ (β, λ) , 0 ≤β < 1 and λ ≥1. Then |a2| ≤min (r 2 (1 −β) 2λ + 1 , 2 (1 −β) λ + 1 ) and |a3| ≤2 (1 −β) 2λ + 1 . If we choose λ = µ = 1 in first parts of assertions (3.3) and (3.4) of Theorem 3.1, we have the following corollary.

Corollary 3.3. Let f(z) given by (1.1) be in the class N 1 Σ (β, 1) , 0 ≤β < 1. Then |a2| ≤ ( q 2(1−β) 3 ; 0 ≤β < 1 3 1 −β 1 3 ≤β < 1 and |a3| ≤2 (1 −β) 3 .

Corollary 3.4. Let f(z) given by (1.1) be in the class S∗ Σ (β) , 0 ≤β < 1. Then |a2| ≤ p 2 (1 −β) and |a3| ≤  2 (1 −β) ; 0 ≤β < 3 4 (1 −β)(5 −4β); 3 4 ≤β < 1 .