Radii of covering disks for locally univalent harmonic mappings

TL;DR

This paper studies how the size of the largest univalent disks on the manifold changes when replacing analytic functions with harmonic mappings, providing bounds and estimates for various classes of harmonic functions.

Contribution

It introduces sharp bounds for the ratio of covering disk radii between harmonic and analytic mappings, and estimates the convexity radius for specific harmonic function families.

Findings

Sharp bounds for the ratio d_f(z_0)/d_h(z_0) for harmonic vs. analytic mappings.

Estimates on the convexity radius for certain harmonic function families.

Analysis of Q-quasiconformal harmonic mappings on the unit disk.

Abstract

For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex plane onto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largest univalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). The main aim of the present article is to investigate how the radius $d_h(z_0)$ varies when the analytic function $h$ is replaced by a sense-preserving harmonic function $f=h+\overline{g}$. The main result includes sharp upper and lower bounds for the quotient $d_f(z_0)/d_h(z_0)$, especially, for a family of locally univalent $Q$-quasiconformal harmonic mappings $f=h+\overline{g}$ on $|z|<1$. In addition, estimate on the radius of the disk of convexity of functions belonging to certain linear invariant families of locally univalent $Q$-quasiconformal harmonic mappings of order $\alpha$ is obtained.