A certain subclass of meromorphically q-starlike functions associated with the Janowski functions

Theorem 1 A function f ∈M of the form given by (1.5) is in the class MS∗ q[A,B] if it satisfies the following condition: ∞  n=1 Λ(n,A,B,q)|an| ≤Υ (A,B,q), (2.1) where Λ(n,A,B,q) = 2

Theorem 1 is thus completed. □ It is easy to deduce the following consequence of Theorem 1.

Corollary 1 If a function f ∈M of the form given by (1.5) is in the class MS∗ q[A,B], then an ≤Υ (A,B,q) Λ(n,A,B,q) (n ∈N) (2.5) with equality for each n, with the function of the form fn(z) = 1 z + Υ (A,B,q) Λ(n,A,B,q)zn, where Υ (A,B,q) and Λ(n,A,B,q) are given by (2.2) and (2.3) respectively. 3 Distortion inequalities

Theorem 2 If f ∈MS∗ q[A,B], then 1 r – Υ (A,B,q) Λ(1,A,B,q)r ≤ f (z) ≤1 r + Υ (A,B,q) Λ(1,A,B,q)r

Theorem 3 If f ∈MS∗ q[A,B], then 1 r2 – 2Υ (A,B,q) Λ(1,A,B,q) ≤ f ′(z) ≤1 r2 + 2Υ (A,B,q) Λ(1,A,B,q)

Theorem 4 If f of the form (1.5) satisfies condition (2.1), then ℜ

Theorem 4. □ We next turn to ratios involving derivatives.

Theorem 5 If f of the form (1.1) satisfies condition (2.1), then ℜ

Theorem 6 Let the function f given by (1.5) be in the class MS∗ q[A,B]. Then, if inf n≥1  (1 – α)Λ(n,A,B,q) (n + 2 – α)Υ (A,B,q)  1 n+1 = r is positive, then f is meromorphically starlike of order α in |z| ≤r, where Λ(n,A,B,q) and Υ (A,B,q) are given by (2.2) and (2.3) respectively.