On distortion of quasiregular mappings of the upper half plane

TL;DR

The paper addresses distortion of the hyperbolic-type metric $h_{\mathbb{H}^2,c}$ under $K$-quasiregular maps of the upper half-plane. It combines the Schwarz lemma for quasiregular mappings with a new Bernoulli-type inequality and elliptic-integral-based comparison functions to derive a sharp distortion bound, introducing an explicit constant $\lambda(K) \in [1, e^{\pi(K-1/K)})$. The main result states that $h_{\mathbb{H}^2,c}(f(x),f(y)) \le \lambda(K)^{1/2}\; K^{1+c}\; \max\{ h_{\mathbb{H}^2,c}(x,y)^{1/K}, h_{\mathbb{H}^2,c}(x,y) \}$ for all $x,y$ in the half-plane and $c\ge1$, with the proof relying on a Schwarz-type inequality for quasiregular mappings and a sharp Bernoulli-type inequality. The work extends understanding of distortion phenomena for hyperbolic-type metrics in geometric function theory and yields explicit, potentially sharp constants for the quasiregular setting, including the analytic case $K=1$.

Abstract

We prove a sharp result for the distortion of a hyperbolic type metric under $K$-quasiregular mappings of the upper half plane. The proof makes use of a new kind of Bernoulli inequality and the Schwarz lemma for quasiregular mappings.

Theorems & Definitions (18)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.7
• proof
• Lemma 2.8
• proof
• Lemma 2.11
• proof
• Theorem 2.14