On a subclass of close-to-convex harmonic mappings

TL;DR

This paper introduces a new subclass of close-to-convex harmonic mappings defined by a specific inequality, proves their close-to-convexity under certain conditions, and explores their coefficient bounds, growth, convolution properties, and applications.

Contribution

It defines a novel class of harmonic mappings with a specific inequality, proving close-to-convexity and deriving key properties and applications.

Findings

Functions are close-to-convex for certain parameter ranges.

Coefficient bounds and growth estimates are established.

Constructs harmonic univalent polynomials within the class.

Abstract

For $\alpha > -1$ and $\beta >0, $ let $\mathcal{B}_{\mathcal{H}}^0(\alpha, \beta)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+\alpha(h'(z)-1)|\leq \beta-|zg''(z)+\alpha g'(z)|.$ First, we establish that each function belonging to this class is close-to-convex in the open unit disk if $\beta \in (0, 1+\alpha]$. Next, we obtain coefficient bounds, growth estimates and convolution properties. We end the paper with applications and will construct harmonic univalent polynomials belonging to this class.