Conformally invariant metrics and lack of Hölder continuity

The modulus metric between two points in a subdomain of Rn, n ≥ 2, is defined in terms of moduli of curve families joining the boundary of the domain with a continuum connecting the two points. This metric is one of the conformally invariant hyperbolictype metrics that have become a standard tool in geometric function theory. We prove that the modulus metric is not Hölder continuous with respect to the hyperbolic metric.