Convolution conditions for two subclasses of analytic functions defined by Jackson

Proposition 2  If f ∈A, then and

Theorem 3  A function f of the form (1) in the class S∗ ,ζ(A, B) if and only if where C = Cθ = e−iθ+A A−B , θ ∈[0, 2π).

Corollary 4  A function f of the form (1) in the class S∗ ζ (A, B) if and only if where C = Cθ = e−iθ+A A−B , θ ∈[0, 2π).

Corollary 6  A function f  of the form (1) is in the class Cζ(A, B) if and only if 1 z

Theorem 3, we obtain the result obtained by Lashin [16]. 2 Letting ζ →1−1, A = [b(1 + m) −m] and B = −m with m = 1 −1 M and M > 1 2 in Corollaries 4 and 6, respectively, we obtain the results obtained by El-Ashwah [26,

Theorem 2.1 and Theorem 2.4]. 3 Taking ζ →1−1, A = 1 −2α, B = −1 and eiθ = x(|x| = 1) in Corollaries 4 and 6, we obtain the results obtained by Silverman et al. [23, Theorems 1,2]. 4 Taking ζ →1−1and eiθ = x(|x| = 1) in Corollaries 4 and 6, respectively, we obtain the results obtained by Padmanabhan and Ganesan [27, Theorem 1,2].

Theorem 9  A necessary and sufficient condition for the function f of the form (1) to be in the class S∗ ,ζ(n, A, B) is for all θ ∈[0, 2π) and z ∈U.

Theorem 10  If the function f defined by (1) satisfies the inequality then f (z) ∈S∗ ,ζ(n, A, B).

Theorem 11  S∗ ,ζ(n + 1, A, B) ⊂S∗ ,ζ(n, A, B).

Definition 1  For f ∈A , we say f belongs to class S∗ ,ζ(A, B) , if and only if where 0 < ζ < 1, 0 ≤ ≤1, −1 ≤B < A ≤1, Dζ is Jackson q-derivative with q = ζ and ≺ denotes the usual subordination (see [9–11]). It is noticed that, by giving specific values to A, B and  we obtain the following important subclasses studied by various authors in earlier works: 1. S∗ 0,q(1 −2α, −1) ≡S∗ q(α) and S∗ 1,q(1 −2α, −1) ≡Cq(α) are, respectively, the classes