An Alternative Approach to Convolutions of Harmonic Mappings

TL;DR

This paper introduces a computationally efficient alternative method for studying convolutions of univalent harmonic mappings, enabling more general results beyond previous approaches.

Contribution

It presents a novel approach that simplifies analysis and extends known results in the convolution of harmonic mappings, reducing computational complexity.

Findings

More general convolution results proved

Reduced computational effort demonstrated

Earlier specific results generalized

Abstract

Convolutions or Hadamard products of analytic functions is a well explored area of research and many nice results are available in literature. On the other hand, very little is known in general about the convolutions of univalent harmonic mappings. So, researchers started exploring properties of convolutions of some specific univalent harmonic mappings and while doing so, they have mostly used well known Cohn's rule or/and Schur-Cohn's algorithm, which involves computations that are very cumbersome. The main objective of this article is to present an alternative approach, which requires very less computational efforts and allows us to prove more general results. Most of the earlier known results follow as particular cases of the results proved herein.