A Class of k-Symmetric Harmonic Functions Involving a Certain q-Derivative Operator

Hari M Srivastava,Nazar Khan,Qazi Zahoor Ahmad,Shahid Khan,Bilal Khan

doi:10.3390/math9151812

Abstract

In this paper, we introduce a new class of harmonic univalent functions with respect to k-symmetric points by using a newly-defined q-analog of the derivative operator for complex harmonic functions. For this harmonic univalent function class, we derive a sufficient condition, a representation theorem, and a distortion theorem. We also apply a generalized q-Bernardi–Libera–Livingston integral operator to examine the closure properties and coefficient bounds. Furthermore, we highlight some known consequences of our main results. In the concluding part of the article, we have finally reiterated the well-demonstrated fact that the results presented in this article can easily be rewritten as the so-called (p,q)-variations by making some straightforward simplifications, and it will be an inconsequential exercise, simply because the additional parameter p is obviously unnecessary.

Highlights

We define the q-analog of the derivative operator for the harmonic function f = h + g given by (1) as follows: σ,s σ,s σ,s
The theory of the basic calculus has been applicable in many areas of mathematics and physics such as fractional calculus and quantum physics as described in Srivastava’s recently-published survey-cum-expository review article [12]
Researches on the q-calculus in connection with geometric function theory and, especially, harmonic univalent functions are fairly recent and not much has been published on this topic

Summary

Introduction

A normalized univalent analytic function f is said to be starlike with respect to symmetrical points in U if it satisfies the following condition: Ahuja and Jahangiri [10] discussed the class SH(α) of complex-valued and sense-preserving harmonic univalent functions f of the form (1) and satisfying the following condition: Jahangiri [38] applied certain q-operators to complex harmonic functions and obtained sharp coefficient bounds, distortion theorems, and covering results.

Results

Conclusion