Inequalities for the inner radii of nonorevlapping domains

Abstract

UDC 517.54
We consider the problem of maximum of the functional $$ r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right), $$ where $B_{0},\ldots,B_{n}$, $n\geq 2$, are pairwise disjoint domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$, $k=\overline{1,n},$ and $\gamma\in (0, n]$ ($r(B,a)$ is the inner radius of the domain $B\subset\overline{\mathbb{C}}$ with respect to $a$). Show that it attains its maximum at a configuration of domains $B_{k}$ and points $a_{k}$ possessing rotational $n$-symmetry. This problem was solved by Dubinin for $\gamma=1$ and by Kuz’mina for $0<\gamma< 1$. Later, Kovalev solved this problem for $n\geqslant5$ under an additional assumption that the angles between neighboring linear segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$. We generalize this problem to the case of arbitrary locations of the systems of points in the complex plane and obtain some estimates for the functional for all $n$ and $\gamma\in (1, n]$.