Mathematical Communications 13(2008), 311-319

Theorem 1. [6]If −1 ≤B < A ≤1, β ∈C \ {0} and the complex number η satisfies Re(η) ≥−β(1 −A)/(1 −B), then the differential equation q(z) + zq ′(z) βq(z) + η = 1 + Az 1 + Bz , ∀z ∈U has a univalent solution in U given by q(z) =    zβ+η (1+Bz)β(A−B)/(B) β

Theorem 2. [7] Let the function q be univalent in the open unit disc U and θ and φ be analytic in a domain D containing q(U) with φ(w) ̸= 0 when w ∈q(U). set Q(z) = zq ′(z)φ(q(z)), h(z) = θ(q(z)) + Q(z). Suppose that 1. Q is starlike univalent in U, and 2. Re  zh ′(z) Q(z)  > 0 for z ∈U. If θ(p(z)) + zp ′(z)φ(p(z)) ≺θ(q(z)) + zq

Theorem 3. [4] Let the function q be univalent in the open unit disc U and ϑ and φ be analytic in a domain D containing q(U). Suppose that 1. Re  ϑ ′(q(z)) φ(q(z))  > 0 for z ∈U and 2. zq ′(z)φ(q(z)) is starlike univalent in U. If p ∈H[q(0), 1] ∩Q, with p(U) ⊆D, and ϑ(p(z)) + zp ′(z)φ(p(z)) is univalent in U and ϑ(q(z)) + zq

Theorem 4. Let δ be a non-zero complex number and let the function q(z) be analytic and univalent in U such that q(z) ̸= 0, ∀z ∈U. Suppose that zq ′(z) q(z) is starlike univalent in U. Let Re 1 δ q(z) + 1 + zq ′′(z) q ′(z) −zq ′(z) q(z)

Corollary 1. Let δ be a non-zero complex number and assume that (9) holds. If f ∈A and Ψm λ (α1, β1, δ, f)(z) ≺1 + Az 1 + Bz + δ (A −B)z (1 + Az)(1 + Bz)

Corollary 2. Let δ be a non-zero complex number and assume that (9) holds. If f ∈A and Ψm λ (α1, β1, δ, f)(z) ≺ 1 + z 1 −z µ + 2δµ z (1 −z2), where Ψm λ (α1, β1, δ, f) is defined as in (10), then for 0 ≤γ ≤1, (Dm λ (α1, β1)f(z)) ′

Theorem 5. Let −1 ≤B < A ≤1 , 0 ≤γ ≤1 and δ ∈C\Z− 0 (Z− 0 = {0, −1, −2, . . . , }). If (Dm λ (α1, β1)f(z)) ′ z Dm λ (α1, β1)f(z) 1+γ (12) +δ z(Dm

Corollary 3. Let −1 ≤B < A ≤1 and δ ∈C\Z− 0 (Z− 0 = {0, −1, −2, . . . , }). If z2f ′(z)  f(z) 2 + δ zf ′′(z) f ′(z) + 2  1 −zf

Theorem 6. Let δ be a non-zero complex number and let q be analytic and univalent in U such that q(z) ̸= 0 and zq ′(z) q(z) starlike univalent in U. Further, let us assume that Re q(z) δ  > 0. (15) If f ∈A, 0 ̸= (Dm

Theorem 7. Let δ be a non-zero complex number and let q1 and q2 be univalent in U such that q1(z) ̸= 0 and q2(z) ̸= 0, ∀z ∈U with zq ′ 1(z) q1(z) and zq ′ 2(z) q2(z) being starlike univalent. Suppose that q1 satisfies (15) and q2 satisfies (9). If f ∈A, (Dm λ (α1, β1)f(z)) ′ z Dm λ (α1, β1)f(z)

Corollary 4. [10]Let δ be a non-zero complex number and let q1 and q2 be univalent in U such that q1(z) ̸= 0 and q2(z) ̸= 0, (z ∈U) with zq ′ 1(z) q1(z) and zq ′ 2(z) q2(z) being starlike univalent. Suppose that q1 satisfies (15) and q2 satisfies (9). If f ∈A, z2f ′(z)