Analogs of the Ikoma–Schwartz lemma and Liouville theorem for mappings with unbounded characteristic

Abstract

We obtain results on the local behavior of open discrete mappings $ f:D \to {\mathbb{R}^n} $ , n ≥ 2, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known Ikoma growth estimate proved for quasiconformal mappings of the unit ball into itself and of the classic Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analog of the well-known Liouville theorem for analytic functions.