The problem of product of conformal radii of nonoverlapping domains

V. O. Kuznetsov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 12, Tome 212 (1994), pp. 114-128

Résumé

Let $a_1,a_2,a_3,b$ be distinct points in $\overline{\mathbb C}$ and let $\mathcal D$ be the family of all triples of nonoverlapping domains $D_1, D_2,D_3$ in $\overline{\mathbb C}\setminus b$ such that $a_k\in D_k$, $k=1,2,3$. For this family we consider the problem on the maximum of the functional $I=R_1R_2R_3$, where $R_k=R(D_k,a_k)$ is the conformal radius of $D_k$ with respect to $a_k$. Geometrical properties of the extremal triple of domains are described. We prove that the maximum of $I$ monotonically depends on the position of the point $b$ and find the maximum in some special cases. Bibliography: 10 titles.