Some estimation about Tayler-Maclaurin coefficients of generalized subclasses of bi-univalent functions

Lemma 1.3. [9] If h ∈P, then the estimates |cn| ≤2, n = 1, 2, 3, ... are sharp, where P is the family of all functions h which are analytic in U for which h(0) = 1 and Re(h(z)) > 0(z ∈U) where h(z) = 1 + c1z + c2z2 + ..., z ∈U. 2. Coefficient bounds for the function class Sα Σ(τ, δ, λ, γ) In this section, we establish coefficient bounds for the Taylor-Maclaurin coefficients |a2| and |a3| of the function f ∈Sα Σ(τ, δ, λ, γ).

Theorem 2.1. let f(z) defined by (1.1) belonging to the class Sα Σ(τ, δ, λ, γ), then |a2| ≤ 2α|τ| p |2ατΩ+ (1 −α)(1 + δ + 2µ −λ −γλ)2| . (2.1) and |a3| ≤min ( 4α2|τ|2 (1 + δ + 2µ −λ −γλ)2 + 2α|τ| |1 + 2δ + 6µ −λ −2γλ|, 2α|τ|

Theorem 3.1. let f(z) defined by (1.1) belonging to the class SΣ(τ, δ, λ, γ; β), then |a2| ≤min  2(1 −β)|τ| |1 + δ + 2µ −λ −γλ|, s 2|τ|(1 −β) |Ω|  . (3.1) and |a3| ≤min ( 4(1 −β)2|τ|2

Corollary 4.1. Let f(z) given by (1.1) be in the class HΣ(τ, µ, λ, γ; α) then, |a2| ≤ 2α|τ| q |2ατ ˜Ω+ (1 −α)(2 + 2µ −λ −γλ)2| . 6

Corollary 4.2. Let f(z) given by (1.1) be in the class HΣ(τ, µ, λ, γ; β) then, |a2| ≤min  2(1 −β)|τ| |2 + 2µ −λ −γλ|, s 2|τ|(1 −β) | ˜Ω|  . and |a3| ≤min ( 4(1 −β)2|τ|2 |2 + 2µ −λ −γλ|2 +

Corollary 4.3. Let f(z) given by (1.1) be in the class NΣ(β, λ, δ) then, |a2| ≤min  2(1 −β) 1 + λ + 2δ, r 2(1 −β) 1 + 2λ + 6δ , |a3| ≤ 2(1 −β) 1 + 2λ + 6δ.

Corollary 4.4. Let f(z) given by (1.1) be in the class GΣ(α, λ) then, |a2| ≤ 2α (1 −λ) √ 1 + α , |a3| ≤min ( 4α2 (1 −λ)2 + α 1 −λ, 2α (1 −λ)2

Corollary 4.5. Let f(z) given by (1.1) be in the class MΣ(β, λ) then, |a2| ≤ p 2(1 −β) 1 −λ , |a3| ≤min (2(1 −β) (1 −λ)2 , 4(1 −β)2 (1 −λ)2 + 1 −β 1 −λ ) .

Corollary 4.6. Let f(z) given by (1.1) be in the class Sa,1,a Σ (α, λ) then, |a2| ≤ 2α p |2α(λ2 −3λ + 3) + (1 −α)(2 −λ)2| , |a3| ≤min ( 4α2 (2 −λ)2 + 2α 3 −λ, 2α

Corollary 4.7. Let f(z) given by (1.1) be in the class Ma,1,a Σ (β, λ) then, |a2| ≤ r 2(1 −β) λ2 −3λ + 3, |a3| ≤min ( 2(1 −β) λ2 −3λ + 3, 4(1 −β)2 (2 −λ)2 + 2(1 −β) 3 −λ ) .

Corollary 4.8. Let f(z) given by (1.1) be in the class BΣ(α, λ) then, |a2| ≤ 2α q4α(1 + 2λ) + (1 −3α)(1 + λ)2 , |a3| ≤min ( 4α2 (1 + λ)2 + α 1 + 2λ, 2α 1 + 2λ −λ2 ) .

Corollary 4.9. Let f(z) given by (1.1) be in the class NΣ(β, λ) then, |a2| ≤ r 2(1 −β) 1 + 2λ −λ2 , |a3| ≤min ( 2(1 −β) 1 + 2λ −λ2 , 4(1 −β)2 (1 + λ)2 + 1 −β 1 + 2λ ) .

Corollary 4.10. Let f(z) given by (1.1) be in the class BΣ(α, λ) then, |a2| ≤ 2α p |α(1 + 2λ −λ2) + (1 + λ)2 , |a3| ≤ 2α 1 + 2λ.

Corollary 4.11. Let f(z) given by (1.1) be in the class BΣ(β, λ), then |a2| ≤ r 2(1 −β) 1 + 2λ , |a3| ≤2(1 −β) 1 + 2λ .

Corollary 4.12. Let f(z) given by (1.1) be in the class HΣ(α, β) then, |a2| ≤ 2α p |2(2 + α) + 4β(α + β + 2 −αβ)| , |a3| ≤ 2α 3(1 + 2β). 8

Corollary 4.13. Let f(z) given by (1.1) be in the class HΣ(γ, β), then |a2| ≤ s 2(1 −γ) 3(1 + 2γ), |a3| ≤2(1 −γ) 3(1 + 2β).

Corollary 4.14. Let f(z) given by (1.1) be in the class Hα Σ then, |a2| ≤α r 2 2 + α, |a3| ≤2α 3 .

Corollary 4.15. Let f(z) given by (1.1) be in the class HΣ(β), then |a2| ≤ r 2(1 −β) 3 , |a3| ≤2(1 −β) 3 .