Injectivity of sections of close-to-convex harmonic mappings with convex analytic part

TL;DR

This paper establishes distortion theorems, coefficient estimates, and univalence radii for close-to-convex harmonic mappings with convex analytic parts, providing explicit bounds and numerical comparisons.

Contribution

It introduces new sharp coefficient bounds and univalence radii for a family of harmonic mappings with convex analytic parts of order alpha.

Findings

Two-point distortion theorem established

Sharp coefficient estimates derived

Explicit lower bounds for univalence radius obtained

Abstract

In this article, we determine two point distortion theorem and sharp coefficient estimates for the families of close-to-convex harmonic mappings whose analytic part is a convex function of order $\alpha$. By making use of these results, we determine the radius of univalence of sections of these families in terms of zeros of certain equation. Lower bound for the radius of univalence has been obtained explicitly for the case $\alpha = 1/2$. Comparison of radius of univalence of the sections have been shown by providing a table of numerical estimates for the special choices of $\alpha$.