Radius of injectivety for harmonic mappings with fixed analytic part

TL;DR

This paper investigates the radius within which harmonic mappings with a convex and injective analytic part remain injective, expanding understanding of their geometric properties.

Contribution

It provides the radius of injectivity for harmonic mappings with a fixed convex and injective analytic component in the unit disk.

Findings

Determines the radius of injectivity for specified harmonic mappings.

Extends geometric function theory to harmonic mappings with convex analytic parts.

Provides new bounds for injectivity radius in harmonic mapping class.

Abstract

In this paper, we study non sense-preserving harmonic mappings $f=h+\overline{g}$ in $\mathbb{D}$ when its analytic part $h$ is convex and injective in $\mathbb{D}$ and obtain radius of injectivety.