Decompositions into nonoverlapping domains and extremal properties of univalent functions

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.