Decompositions into nonoverlapping domains and extremal properties of univalent functions

A. Yu. Solynin

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 12, Tome 212 (1994), pp. 139-163

Résumé

We apply the method of extremal metrics and certain symmetrization approaches to study problems on conformal mappings of a disk and circular annulus. For instance, we solve the problem on the maximal conformal module in the family of all doubly-connected domains of the form $\overline{\mathbb C}\setminus(E_1\cup E_2)$ with $E_1\cap E_2=\varnothing$, $r_1,r_2\in E_1$, $0\le r_1,r_2\le\infty$, and $\operatorname{diam}E_2\cap\{z\colon|z|<1\}\ge\lambda>0$. This generalizes the classical result by A. Mori. We also give a new solution of a problem by P. M. Tamrazov, which was initially solved by V. A. Shlyk. Some new theorems on the covering of a regular system of $n$ rays are obtained for certain classes of convex mappings. Bibliography: 22 titles.