Sharp estimates of products of inner radii of non-overlapping domains in the complex plane

A. K. Bakhtin ; I. V. Denega

Problemy analiza, Tome 8 (2019) no. 1, pp. 17-31

Résumé

In the paper we study a generalization of the extremal problem of geometric theory of functions of a complex variable on non-overlapping domains with free poles: Fix any $\gamma\in\mathbb{R^{+}}$ and find the maximum (and describe all extremals) of the functional $$ \left[r\left(B_0,0\right)r\left(B_\infty,\infty\right)\right]^{\gamma} \prod\limits_{k=1}^n r\left(B_k,a_k\right), $$ where $n\in \mathbb{N}$, $n\geqslant 2$, $a_{0}=0$, $|a_{k}|=1$, $B_0$, $B_\infty$, $\{B_{k}\}_{k=1}^{n}$ is a system of mutually non-overlapping domains, $a_{k}\in B_{k}\subset\overline{\mathbb{C}}$, $k=\overline{0, n}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, ($r(B,a)$ is an inner radius of the domain $B\subset\overline{\mathbb{C}}$ at $a\in B$). Instead of the classical condition that the poles are on the unit circle, we require that the system of free poles is an $n$-radial system of points normalized by some "control" functional. A partial solution of this problem was is obtained.

Détail

Keywords: inner radius of a domain, non-overlapping domains, radial system of points, separating transformation, quadratic differential, Green's function.