Radius Properties of Harmonic Mappings with Fixed Analytic Part

TL;DR

This paper studies harmonic mappings with a fixed analytic part in the unit disk, establishing sharp radius results for various geometric properties and enhancing existing radius theorems.

Contribution

It provides new sharp radius bounds for harmonic mappings with analytic parts in the Ma and Minda class, extending known results.

Findings

Sharp radius for univalency established

Radius of fully starlikeness determined

Enhanced radius results for harmonic mappings

Abstract

We investigate harmonic mappings $f=h+\bar{g}$ defined in the unit disk, where $g$ and $h$ satisfy certain prescribed conditions and the analytic part $h$ belongs to the Ma and Minda class of starlike functions. Certain sharp radius results for univalency, close-to-convexity, fully starlikeness, fully convexity, and radius of strongly starlikeness are established. We also calculate the radius of uniformly starlikeness and convexity for these functions. Several results enhance the well-known radius results.