Maximum of a conformal invariant in the problem of nonoverlapping domains

S. I. Fedorov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 4, Tome 112 (1981), pp. 172-183

Résumé

The paper is devoted to the well-known circle of problems on the maximum of the products of powers of conformal radii of nonoverlapping domains. Let $a_1,\dots,a_n$ be distinct points of $\mathbb C$ and let $D_1,\dots,D_n$ be a system of simply connected domains in $\overline{\mathbb C}$, pairwise disjoint and such that $a_k\in D_k$, $k=1,\dots,n$. By $R(D_k,a_k)$ we denote the conformal radius of the domain $D_k$ relative to the point $a_k$. One considers the problem on the maximum of the product $$ \prod^n_{k=1}R(D_k,a_k)\Bigl\{\prod_{1\le k<l\le n}|a_k-a_l|\Bigr\}^{-2/(n-1)} $$ in the family of all indicated systems of domains, under the condition that $a_1,\dots,a_n$ runs over all systems of distinct points in $\mathbb C$ ($n\ge4$) and one finds the geometric characteristic of the extremal configurations of this problem in terms of the associated quadratic differential.