The problem of product of conformal radii of nonoverlapping domains

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.