On the problem of extremal partition of the closed plane

One considers the problem of the maximum of the product of powers of conformal radii of nonoverlapping domains in the following formulation. Let $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_m\}$ be systems of distinguished points in $\bar{ \mathbb{C} }$ and let $\alpha=\{\alpha_1,\dots,\alpha_m\}$ be a system of positive numbers. By $M(D_\ell,b_\ell)$ we denote the reduced modulus of the simply connected domain $D_\ell$ relative to the point $b_\ell\in D_\ell$. Find the maximum of the sum
$$ \sum^m_{\ell=1}\alpha^2_\ell M(D_\ell,b_\ell) $$
in the family $D$ of all systems of nonoverlapping simply connected domains $D_j$, $j=1,\dots,m$, satisfying the following condition: the domain $D_j$ does not contain points $b_i\in B$, different from $b_j$, and some collection $A_j$, for each domain, of points from $A$, $U^m_{j=1}A_j=A$. The solution of this problem is obtained by the simultaneous use of the method of variation and of the method of the moduli of families of curves and is given by Theorem 1 of the present paper. As consequences of Theorem 1 one obtains Theorems 2 and 3, strengthening the corresponding results of a previous paper of the author.