Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions

Abstract

In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric q-calculus and the symmetric q-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions SH0˜m,q,A,B. First, we illustrate the necessary and sufficient convolution condition for SH0˜m,q,A,B and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass TSH0˜m,q,A,B. Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of q-starlike and q-convex functions of order α, and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results.