On Mori's theorem for quasiconformal maps in the $n$-space

TL;DR

This paper provides an alternative, quantitatively bounded proof of Mori's theorem for quasiconformal maps in n-space, including explicit bounds and a Schwarz lemma version, supported by experimental comparisons.

Contribution

It introduces a new proof with explicit bounds for Mori's constant and extends the Schwarz lemma to quasiconformal self-maps of the unit disk.

Findings

Quantitative upper bounds for Mori's constant in n-space.

Explicit Schwarz lemma for quasiconformal maps of the unit disk.

Experimental comparison of bounds for Mori's constant in 2D.

Abstract

R. Fehlmann and M. Vuorinen proved in 1988 that Mori's constant $M(n,K)$ for $K$-quasiconformal maps of the unit ball in $\mathbf{R}^n$ onto itself keeping the origin fixed satisfies $M(n,K) \to 1$ when $K\to 1 .$ We give here an alternative proof of this fact, with a quantitative upper bound for the constant in terms of elementary functions. Our proof is based on a refinement of a method due to G.D. Anderson and M. K. Vamanamurthy. We also give an explicit version of the Schwarz lemma for quasiconformal self-maps of the unit disk. Some experimental results are provided to compare the various bounds for the Mori constant when $n=2 .$