On quasiinvariance of harmonic measure and Hayman–Wu theorem

S. Yu. Graf<sup>ab

a</sup> Tver State University, 33 Zheliabova str., Tver, 170100 Russia
^b Petrozavodsk State University, 33 Lenina Ave., Petrozavodsk, 185910 Russia

Abstract: The article is devoted to the definition and properties of the class of diffeomorphisms of the unit disk $\mathbb{D}=\{z: |z|<1\}$ 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 quasiinvariant. Estimates for derivative mappings of this class are obtained. We prove that such mappings are quasiconformal and are also quasiisometries 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 mappings is proved.

Keywords: harmonic measure, quasiconformal mapping, pseudohyperbolic metric, quasiisometry, Hayman–Wu theorem.

UDC: 517.54

Received: 27.01.2023
Revised: 27.01.2023
Accepted: 29.03.2023

DOI: 10.26907/0021-3446-2024-2-22-36

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2024, 68:2, 18–30