q-STARLIKE FUNCTIONS OF ORDER ALPHA

Abstract

For all q ∈ (0, 1) and 0 ≤ α < 1 we define a class of analytic functions, so-called q-starlike functions of order α on the open unit disc D = {z : |z| < 1} . We will study this class of functions and explore some inclusion properties with the well-known class Starlike functions of order α.

On q-Uniformly Mocanu Functions

“…By taking a = 1−α 1−q , b = 1−α 1−q in 4, it has been shown in [17] that f ∈ S * q (α), if and only if,…”

A new class of harmonic functions associated with a (p,q)-Ruscheweyh operator

“…Research work in connection with function theory and q-calculus was first introduced by Ismail et al [8]. Recently, the q-calculus and its generalization called post-quantum calculus also known as the (p, q)-calculus has been involved in the theory of analytic and harmonic functions in the work [1,2,4,6,7,11,13,15,16,18] (see also [3]). Some definitions related to the (p, q)-calculus are as follows: Definition 1.…”

“…Theorem 3. Let f = h + g, where h and g are of the form (18).Then f ∈ TS 0 H (n, p, q, α) if and only if the condition (15) holds.…”

“…where ψ k is given by ( 5) and this proves that the class TS 0 H (n, p, q, α) is locally uniformly bounded. Let f = h + g, where h and g are given by (18). Further, since for t = 1, 2, 3, ..., f t ∈ TS 0 H (n, p, q, α) be of the form (19), by Theorem (3), we have…”

“…Corollary 5. Let f = h + g ∈ TS 0 H (n, p, q, α), where h and g are of the form (18). Then where ψ k is given by (5).…”