Sharp distortion theorems for quasiconformal mappings

Abstract:

Continuing their earlier work on distortion theory, the authors prove some dimension-free distortion theorems for $K$-quasiconformal mappings in ${R^n}$. For example, one of the present results is the following sharp variant of the Schwarz lemma: If $f$ is a $K$-quasiconformal self-mapping of the unit ball ${B^n}$, $n \geqslant 2$, with $f(0) = 0$, then ${4^{1 - {K^2}}}|x{|^K} \leqslant |f(x)| \leqslant {4^{1 - 1/{K^2}}}|x{|^{1/K}}$ for all $x$ in ${B^n}$.