Hyperbolic metric on the strip and the Schwarz lemma for HQR mappings

TL;DR

This paper presents simplified proofs of Schwarz lemmas for harmonic and HQR mappings with strip codomain, utilizing hyperbolic geometry and subordination principles to derive optimal estimates.

Contribution

It introduces new simplified proofs of Schwarz lemmas for harmonic and HQR mappings, including a novel version for real harmonic functions, and establishes optimal bounds using hyperbolic geometry.

Findings

New simplified proofs of Schwarz lemmas for harmonic functions.

Optimal estimates for the modulus of HQR mappings.

Application of hyperbolic geometry of the strip to distortion estimates.

Abstract

We give simple proofs of various versions of the Schwarz lemma for real valued harmonic functions and for holomorphic (more generally harmonic quasi\-re\-gu\-lar, shortly HQR) mappings with the strip codomain. Along the way using the principle of subordination and the corresponding conformal mapping, depicted on the Figure 1, we get a simple proof of a new version of the Schwarz lemma for real valued harmonic functions (see Theorems 4 and 5) and Theorem 6 related to holomorphic mappings. Using the Schwarz-Pick lemma related to distortion for harmonic mappings and the elementary properties of the hyperbolic geometry of the strip we prove Lemma 4, which is a key ingredient in the proof of Theorem 7 which yields optimal estimates for modulus of HQR mappings.