On Quasi-Invariance of Harmonic Measure and Hayman–Wu Theorem

Abstract

This study defines and describes the properties of the class of diffeomorphisms of the unit disk \(\mathbb{D} = \{ z\,:\;|{\kern 1pt} z{\kern 1pt} |\; < 1\} \) on the complex plane \(\mathbb{C}\) for which the harmonic measure of the boundary arcs of the slit disk has a limited distortion (i.e., is quasi-invariant). Estimates for derivative mappings of this class are obtained. We prove that such mappings are quasiconformal and quasi-isometries with respect to the pseudohyperbolic metric. An example of a mapping with the specified property is given. As an application, a generalization of the Hayman–Wu theorem to this class of such mappings is proved.