Geometric versions of Schwarz’s lemma for quasiregular mappings

Abstract

We prove monotonicity and distortion theorems for quasiregular mappings defined on the unit ball B n of ℝ n . Let K I (f) be the inner dilatation of f and let α = K I (f) 1/(1―n) . Let m n denote n-dimensional Lebesgue measure and c n be the reduced conformal modulus in ℝ n . We prove that the functions r ―nα m n (f(rB n ) and r ―α c n (f(rB n )) are increasing for 0 < r < 1. These results can be viewed as variants of the classical Schwarz lemma and as generalizations of recent results by Burckel et al. for holomorphic functions in the unit disk.