2010/12/20 by Dimitrios Betsakos · 2 citations
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Algebraic and Geometric Analysis
paper · pdf · doi:10.1090/s0002-9939-2010-10604-4
We prove monotonicity and distortion theorems for quasiregular mappings defined on the unit ball \mathbb Bn of \mathbb 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 \mathbb Rn. We prove that the functions r-nα mn(f(r\mathbb Bn)) and r-α cn(f(r\mathbb Bn)) 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.