Certain geometric properties of close-to-convex harmonic mappings

TL;DR

This paper introduces a new family of sense-preserving harmonic mappings in the unit disk, demonstrating their close-to-convexity and analyzing properties like coefficient bounds, growth, and convexity radius.

Contribution

It defines a novel family of harmonic mappings, proves their close-to-convexity, and explores their fundamental properties and specific polynomial examples.

Findings

Functions are close-to-convex within the family.

Coefficient bounds and growth estimates are established.

Convexity radius for the family is determined.

Abstract

In this article, we introduce a new family of sense preserving harmonic mappings f in the open unit disk and prove that functions in this family are close-to-convex. We give some basic properties such as coefficient bounds, growth estimates, convolution and determine the radius of convexity for the functions belonging to this family. In addition, we construct certain harmonic univalent polynomials belonging to this family.