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Let D be a bounded domain in Rn, n ≥ 2, and let f be a continuous mapping of &Dmacr; into Rn which is quasiconformal in D. Suppose that jf(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 max{ω(|x - y|), |x - y|α} for all x; y 2 D, where α = KI (f)1/(1-n). [ABSTRACT FROM AUTHOR]