Hilbert metric and H\"older continuity

TL;DR

This paper investigates the Hilbert metric in the unit disk, deriving formulas and applying them to analyze the H"older continuity of quasiregular mappings between the disk and convex domains, while proposing open problems.

Contribution

It provides new formulas for the Hilbert metric in the unit disk and studies the H"older continuity of quasiregular mappings with respect to these metrics.

Findings

Formulas for the Hilbert metric in the unit disk

H"older continuity results for quasiregular mappings

Open problems in the area

Abstract

We prove several formulas for the Hilbert metric in the unit disk and apply these results to study quasiregular mappings of the unit disk $\mathbb{B}^2$ onto a bounded convex domain $D$. The main result deals with the H\"older continuity of these mappings with respect to Hilbert metrics of $\mathbb{B}^2$ and $D$. Also several open problems are formulated.