On the problem on maximizing the product of powers of conformal radii of nonverlapping domians

E. G. Emel'yanov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 18, Tome 286 (2002), pp. 103-114

Résumé

A sharp estimate of the product $$ \prod^4_{k=1}R^{\alpha^2_k}(D_k,b_k) $$ (as usual,$R(D,b)$ denotes the conformal radius of a domian $D$ with respect to a point $b\in D$) in the family of all quadruples of nonoverlapping simply connected domians $\{D_k\}$, $b_k\in D_k$, $k=1,\dots,4$, is obtained. Here, $\{b_1,\dots,b_4\}$ are four arbitrary distinct points on $\overline{\mathbb C}$, $\alpha_1=\alpha_2=1$, $\alpha_3=\alpha_4=\alpha$, and $\alpha$ is an arbitrary positive number. The proof involves the solution of the problem on maximizing a certain conformal invariant, which is related to the problem under consideration.