Sharp Lipschitz constants for the distance ratio metric

TL;DR

This paper investigates how certain mappings in Euclidean space, including Möbius transformations and polynomials, expand or contract the distance ratio metric, providing sharp Lipschitz constants for these transformations.

Contribution

It derives the exact Lipschitz constants for Möbius transformations of the unit ball and polynomials of the unit disk, advancing understanding of their metric properties.

Findings

Sharp Lipschitz constants for Möbius transformations

Sharp Lipschitz constants for polynomials

Insights into metric distortion under these mappings

Abstract

We study expansion/contraction properties of some common classes of mappings of the Euclidean space ${\mathbb R}^n, n\ge 2\,,$ with respect to the distance ratio metric. The first main case is the behavior of M\"obius transformations of the unit ball in ${\mathbb R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained.