Harmonic mappings, univalence criteria and a theorem of Lehtinen

Abstract

The harmonic inner radius $σ_H(Ω)$ of a planar domain $Ω$ is the largest constant with which a univalence criterion via the Schwarzian derivative holds for harmonic mappings. We show that $σ_H(Ω)\leqσ_H(\mathbb{D})\leq 3/2$ for the unit disk $\mathbb{D}$ and for every domain $Ω$ 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.

Figures (2)

• Figure 1: The range of $f_1$
• Figure 2: The range of $f_{1,5}$

Theorems & Definitions (12)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Conjecture 1
• Conjecture 2
• Lemma 1
• proof
• Theorem 2
• proof