Applications of a q-Differential Operator to a Class of Harmonic Mappings Defined by q-Mittag–Leffler Functions

Abstract

Many diverse subclasses of analytic functions, q-starlike functions, and symmetric q-starlike functions have been studied and analyzed by using q-analogous values of integral and derivative operators. In this paper, we define a q-analogous value of differential operators for harmonic functions with the help of basic concepts of quantum (q-) calculus operator theory; and introduce a new subclass of harmonic functions associated with the Janowski and q-Mittag–Leffler functions. We obtain several useful properties of the new class, such as necessary and sufficient conditions, criteria for analyticity, compactness, convexity, extreme points, radii of starlikeness, radii of convexity, distortion bounds, and integral mean inequality. Furthermore, we discuss some applications of this study in the form of some results and examples and highlight some known corollaries for verifying our main results presented in this investigation. Finally, the conclusion section summarizes the fact about the (p,q)-variations.

Certain subclasses of harmonic functions involving $q-$Mittag-Leffler Function

“…Stirred by the prior works (see [7,11,17,18,19,20,22,27,28,29,33]) on the subject of harmonic functions, in this paper we obtain a sufficiency criteria for functions h given by (1.2) to be in the class HS q,σ ϑ,ρ (µ, ξ). It is shown that this criteria is also necessary for hHS q,σ ϑ,ρ (µ, ξ).…”

Analytic Functions Related to a Balloon-Shaped Domain

“…Let (c, q, y) ∈ [0, 2) × [0, 1) × (0, 1). By differentiating G(c, q, y) with respect to y, we obtain 2 4 − c 2 q(8q + 66) + 5c 2 (16q + 1)…”

“…Using the concept of subordination, many subclasses have been defined and studied, such as S * , C, K and R of starlike, convex, close to convex, and functions with bounded turnings, respectively. See [2][3][4][5][6] for the new results about more subclasses.…”