Sharp distortion theorems for quasiconformal mappings

Abstract

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