Coefficient estimates for some families of bi-bazilevic functions of the ma-minda defined by convolution

Lemma 1.6. [31] If p ∈P, then |pk| ≤2, for each k, where P is the family of all functions p analytic in U for which Re{p(z)} > 0, then h(z) = 1 + p1z + p2z2 + p3z3 + ... , z ∈U. We begin by finding the estimates on the coefficients |a2| and |a2| for functions in the class Bk,α,β,δ,λ Σ (γ, φ). 2. COEFFICIENT BOUNDS FOR THE FUNCTION CLASS Bk,α,β,δ,λ Σ (γ, φ)

Theorem 2.1. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (γ, φ). Then |a2| ≤ B1 √2B1 rB2 1  2(γ + 2)Υk 3C(δ, 3) + (γ −1)(γ + 2)  Υk 2C(δ, 2) 2

Corollary 2.2. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (φ). Then |a2| ≤ B1 √B1 rB2 1  2Υk 3C(δ, 3) −  Υk 2C(δ, 2) 2

Corollary 2.3. Let the function f(z) given by (1) be in the class Hk,α,β,δ,λ Σ (1, φ). Then |a2| ≤ B1 √B1 r3B2 1Υk 3C(δ, 3) −4(B2 −B1)  Υk 2C(δ, 2) 2 and |a3| ≤

Corollary 2.4. Let the function f(z) given by (1) be in the class Bα,β,δ,λ Σ (φ). Then |a2| ≤ B1 √B1 qB2 1 2C(δ, 3) −[C(δ, 2)]2
−(B2 −B1) [C(δ, 2)]2 and |a3| ≤ B1 2C(δ, 3) + 

Corollary 2.5. Let the function f(z) given by (1) be in the class Hα,β,δ,λ Σ (1, φ). Then |a2| ≤ B1 √B1 q3B2 1C(δ, 3) −4(B2 −B1) [C(δ, 2)]2 and |a3| ≤ B1 3C(δ, 3) +  B1 2C(δ, 2)

Corollary 2.6. Let the function f(z) given by (1) be in the class Bk,α,β,λ Σ (φ). Then |a2| ≤ B1 √B1 qB2 1 2Υk 3 −[Υk 2]2
−(B2 −B1)[Υk 2]2 and |a3| ≤B1

Corollary 2.7. Let the function f(z) given by (1) be in the class Hk,α,β,λ Σ (1, φ). Then |a2| ≤ B1 √B1 q3B2 1Υk 3 −4(B2 −B1[Υk 2]2 and |a3| ≤B1 3Υk 3 +

Corollary 2.8. Let the function f(z) given by (1) be in the class Bα,β,λ Σ (φ). Then |a2| ≤ B1 √B1 p |B2 1 −(B2 −B1)| and |a3| ≤B1 2 + B2 1. 7

Corollary 2.9. Let the function f(z) given by (1) be in the class Hα,β,λ Σ (1, φ). Then |a2| ≤ B1 √B1 p |3B2 1 −4(B2 −B1)| and |a3| ≤B1 3 + B1 2 2

Theorem 3.1. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (γ, A, B). Then |a2| ≤ √ 2(A −B) r(A −B)  2(γ + 2)Υk 3C(δ, 3) + (γ −1)(γ + 2)  Υk 2C(δ, 2) 2 −2(B + 1)(γ + 1)2 

Corollary 3.2. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (A, B). Then |a2| ≤ A −B r(A −B)  2Υk 3C(δ, 3) −  Υk 2C(δ, 2) 2 −(B + 1) 

Corollary 3.3. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (1, A, B). Then |a2| ≤ A −B r3(A −B)Υk 3C(δ, 3) −4(B + 1)  Υk 2C(δ, 2) 2 and |a3| ≤ A −B 3Υk

Corollary 3.4. Let the function f(z) given by (1) be in the class Bα,β,δ,λ Σ (A, B). Then |a2| ≤ A −B q(A −B) 2C(δ, 3) −[C(δ, 2)]2
−(B + 1) [C(δ, 2)]2 and |a3| ≤A −B 2C(δ, 3) +  A −B C(δ, 2) 2 .

Corollary 3.5. Let the function f(z) given by (1) be in the class Hα,β,δ,λ Σ (1, A, B). Then |a2| ≤ A −B q3(A −B)C(δ, 3) −4(B + 1) [C(δ, 2)]2 and |a3| ≤A −B 3C(δ, 3) +  A −B 2C(δ, 2) 2 . Putting δ = 0, from Corollaries 3.2 and 3.3, we get the following corollaries.

Corollary 3.6. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (A, B). Then |a2| ≤ A −B q(A −B) 2Υk 3 −[Υk 2]2
−(B + 1)[Υk 2]2 and |a3| ≤A −B 2Υk 3

Corollary 3.7. Let the function f(z) given by (1) be in the class Hk,α,β,λ Σ (1, A, B). Then |a2| ≤ A −B q3(A −B)Υk 3 −4(B + 1)[Υk 2]2 and |a3| ≤A −B 3Υk 3 + A −B 2Υk

Corollary 3.8. Let the function f(z) given by (1) be in the class Bα,β,λ Σ (A, B). Then |a2| ≤ A −B p |(A −B) −(B + 1)| and |a3| ≤A −B 2 + (A −B)2. 9

Corollary 3.9. Let the function f(z) given by (1) be in the class Hα,β,λ Σ (1, A, B). Then |a2| ≤ A −B p |3(A −B)2 −4(B + 1)| and |a3| ≤A −B 3 + A −B 2 2 .

Theorem 3.10. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (γ, ζ). Then |a2| ≤ 2√1 −ζ r  2(γ + 2)Υk 3C(δ, 3) + (γ −1)(γ + 2)  Υk 2C(δ, 2) 2 and |a3| ≤

Corollary 3.11. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (ζ). Then |a2| ≤ p 2(1 −ζ) r  2Υk 3C(δ, 3) −  Υk 2C(δ, 2) 2 and

Corollary 3.12. Let the function f(z) given by (1) be in the class Bk,α,β,δ,λ Σ (1, ζ). Then |a2| ≤ s 2(1 −ζ) 3Υk 3C(δ, 3) and |a3| ≤ 2(1 −ζ) 3Υk 3C(δ, 3) +  (1 −ζ) Υk

Corollary 3.13. Let the function f(z) given by (1) be in the class Bα,β,δ,λ Σ (ζ). Then |a2| ≤ 2(1 −ζ) q2C(δ, 3) −[C(δ, 2)]2
and |a3| ≤1 −ζ C(δ, 3) + (2(1 −ζ)) C(δ, 2) 2 . 10

Corollary 3.14. Let the function f(z) given by (1) be in the class Bα,β,δ,λ Σ (1, ζ). Then |a2| ≤ s 2(1 −ζ) 3C(δ, 3) and |a3| ≤2(1 −ζ) 3C(δ, 3) + (1 −ζ) C(δ, 2) 2 . Putting δ = 0, from Corollaries 3.11 and 3.12, we get the following corollaries.

Corollary 3.15. Let the function f(z) given by (1) be in the class Bk,α,β,λ Σ (ζ). Then |a2| ≤ p 2(1 −ζ) q2Υk 3 −[Υk 2]2
and |a3| ≤(1 −ζ) Υk 3 + 2(1 −ζ)

Corollary 3.16. Let the function f(z) given by (1) be in the class Bk,α,β,λ Σ (1, ζ). Then |a2| ≤ s 2(1 −ζ) 3Υk 3 and |a3| ≤2(1 −ζ) 3Υk 3 + (1 −ζ) Υk

Corollary 3.17. Let the function f(z) given by (1) be in the class Bα,β,λ Σ (ζ). Then |a2| ≤ p 2(1 −ζ) and |a3| ≤(1 −ζ) + 4 [(1 −ζ)]2 .

Corollary 3.18. Let the function f(z) given by (1) be in the class Bα,β,λ Σ (1, ζ). Then |a2| ≤ r 2(1 −ζ) 3 and |a3| ≤2(1 −ζ) 3 + [1 −ζ]2 . Concluding Remarks: By specializing the parameters of operator, various other interesting corollaries and consequences of our main results (which are asserted by Theorem 2.1 above) can be derived easily. The details involved shall be left as an exercise for the interested readers. 11