Harmonic mappings, univalence criteria and a theorem of Lehtinen

TL;DR

This paper investigates the harmonic inner radius and univalence criteria for harmonic mappings, establishing bounds analogous to Lehtinen's theorem for holomorphic functions, and introduces new criteria for harmonic univalence.

Contribution

It extends Lehtinen's theorem to harmonic mappings, providing bounds on the harmonic inner radius and new univalence criteria specific to harmonic functions.

Findings

The harmonic inner radius $\sigma_H(\Omega)$ is bounded above by 3/2 for domains omitting an open set.

Established that $\sigma_H(\Omega)\leq\sigma_H(\mathbb{D})$ for the unit disk.

Provided two new univalence criteria for harmonic mappings.

Abstract

The harmonic inner radius $\sigma_H(\Omega)$ of a planar domain $\Omega$ is the largest constant with which a univalence criterion via the Schwarzian derivative holds for harmonic mappings. We show that $\sigma_H(\Omega)\leq\sigma_H(\mathbb{D})\leq 3/2$ for the unit disk $\mathbb{D}$ and for every domain $\Omega$ that omits an open set. This is an analogue of a theorem of Lehtinen in the setting of holomorphic functions. We provide two related univalence criteria for harmonic mappings.