Harmonic mappings, univalence criteria and a theorem of Lehtinen

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.