The fundamental result of the paper is the following. Theorem: Let Γ be a k-quasiconformal Jordan curve and let ⌊ be another Jordan curve (not necessarily quasiconformal). Assume that f maps conformally ext ⌊ onto ext Γ, f(∞)=∞, f′(∞)>0. We assume that there exists a homeomorphism γ between ⌊ and Γ such that[Figure not available: see fulltext.] Then there exist numbers α=α(k)>0 and A=A(k), such that {divides}f(γ(ζ))-ζ{divides}≤ Aε^α, ζεΓ.