On the Distortion Theorem for Quasiconformal Mappings with Fixed Boundary Values

AbstractFor a planar domain Ω with at least three boundary points and ρΩ the hyperbolic metric of Ω with constant curvature −1, G. J. Martin poses a problem that asks, if f is a K-quasiconformal self-homeomorphism of Ω with boundary values given by the identity mapping, whether ρΩ(z,f(z))≤logK holds for z,f(z)∈Ω. In this note, we give a negative answer to this question