On quasiinvariance of harmonic measure and Hayman-Wu theorem

Abstract

The article is devoted to the definition and properties of the class of diffeomorphisms ofthe unit disk D = { z : | z| < 1} on the complex plane C for which the harmonic measure of theboundary arcs of the slit disk has a limited distortion, i.e. is quasiinvariant. Estimates for derivativemappings of this class are obtained. We prove that such mappings are quasiconformal and are alsoquasiisometries with respect to the pseudohyperbolic metric. An example of a mapping with thespecified property is given. As an application, a generalization of the Hayman–Wu theorem to thisclass of mappings is proved.