Quantum approach on linear combination of harmonic univalent mappings

TL;DR

This paper explores how to use q-calculus operators to determine conditions under which linear combinations of harmonic univalent mappings are univalent and convex in the real axis direction, supported by illustrative examples.

Contribution

It introduces new sufficient conditions for linear combinations of harmonic mappings to be univalent and convex using q-calculus techniques.

Findings

Derived conditions for univalence and convexity of linear combinations

Provided examples illustrating the main results

Extended understanding of harmonic mappings via q-calculus

Abstract

In this paper using $q$ calculus operator we obtain some sufficient conditions on $f_1$ and $f_2$ so that their linear combination $% f=tf_{1}+(1-t)f_{2},\ t\in \left[ 0,1\right] $, is univalent and convex in the direction of the real axis. Some examples are also illustrated to support our main results.