Geometric versions of Schwarz’s lemma for quasiregular mappings

Betsakos, Dimitrios

doi:10.1090/S0002-9939-2010-10604-4

Geometric versions of Schwarz’s lemma for quasiregular mappings
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by Dimitrios Betsakos PDF

Proc. Amer. Math. Soc. 139 (2011), 1397-1407 Request permission

Abstract:

We prove monotonicity and distortion theorems for quasiregular mappings defined on the unit ball $\mathbb {B}^n$ of $\mathbb R^n$. Let $K_I(f)$ be the inner dilatation of $f$ and let $\alpha =K_I(f)^{1/(1-n)}$. Let $m_n$ denote $n$-dimensional Lebesgue measure and $c_n$ be the reduced conformal modulus in $\mathbb R^n$. We prove that the functions $r^{-n\alpha }m_n(f(r\mathbb {B}^n))$ and $r^{-\alpha }c_n(f(r\mathbb {B}^n))$ 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.

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