Sharp Lipschitz Constants for the Distance Ratio Metric

Abstract

We study expansion/contraction properties of some common classes of mappings of the Euclidean space $\mathsf{R}^n$, $n\ge 2$, with respect to the distance ratio metric. The first main case is the behavior of Möbius transformations of the unit ball in $\mathsf{R}^n$ 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.