GEOMETRIC VERSIONS OF SCHWARZ’S LEMMA FOR QUASIREGULAR MAPPINGS

Abstract. We prove monotonicity and distortion theorems for quasireg-ular mappings defined on the unit ball Bn of Rn. Let KI(f) be the inner dilatation of f and let α = KI(f) 1/(1−n). Let mn denote n-dimensional Lebesgue measure and cn be the reduced conformal modulus in Rn. We prove that the functions r−nαmn(f(rBn)) and r−αcn(f(rBn)) 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. [4] for holomorphic functions in the unit disk. 1