New Subclass of Analytic Function Related with Generalized Conic Domain Associated with q− Differential Operator

Abstract

The quantum (or q -) calculus is widely applied in various operators which include the q -difference ( q -derivative) operator, and this operator plays an important role in geometric function theory (GFT) as well as in the theory of hypergeometric series. In our present investigation, we introduce and study q -differential operator associated with q -Mittag–Leffler function which is an extension of the Salagean q -differential operator. By using this newly defined operator, we define a new subclass of analytic function and studied certain subclass of analytic function in generalized conic domain Ω k , q , γ . For this class, we investigate structural formula, coefficient estimates, sufficient condition, Fekete–Szegö problem, and also some subordination results.

Partial Sums of the Normalized Le Roy-Type Mittag-Leffler Function

“…Geometric properties, including starlikeness, convexity and close-to-convexity, of the Mittag-Leffler function E α,β (z) were investigated by Bansal and Prajapat in [2] and by Srivastava and Bansal (see [3]). In fact, the generalized Mittag-Leffler function E α,β (z) and its extensions and generalizations continue to be used in many different contexts in geometric function theory (see [4][5][6][7][8][9]).…”