A simple proof of Dubinin's theorem

A. E. Fryntov

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 2 (1995) no. 3, pp. 347-355

Résumé

Let $\Omega$ be a domain formed by removing $n$ radial segments connecting the circles $\{z:| z |=r_0\}$ and $\{z:|z|=1\}$ from the unit disk $\mathbf D$. Let $\Omega_0$ be a domain of the same type which is invariant with respect to rotation by the angle $2\pi/n$. If $\omega(z)$ and $\omega_0(z)$ are the harmonic measures of the unit circle with respect to these domains, then the inequality $$\omega_0\geq\omega_0(0),$$ holds, and the equality is possible only if the domain $\Omega$ coincides with $\Omega_0$ up to rotation. This proposition is known as the Gonchar problem which has been proved by Dubinin. The aim of this paper is to give a more simple proof of this theorem.