Boundary modulus of continuity and quasiconformal mappings

Abstract

Let D be a bounded domain in R n, n ≥ 2, and let f be a continuous mapping of D into R n which is quasiconformal in D. Suppose that |f(x) - f(y)| ≤ ω(|x - y|) for all x and y in ∂D, where ω is a non-negative non-decreasing function satisfying ω(2t) ≤ 2ω(t) for t ≥ 0. We prove, with an additional growth condition on ω, that |f(x) - f(y)| ≤ C maxf{ω(|x - y|); |x - y| α} for all x; y ∈ D, where α = K I(f) 1/(1-n).