Extremal problems of nonoverlapping domains with free poles on a circle
Let α_1, α_2 > 0 and let r(B, a) be the interior radius of the domain B lying in the extended complex plane ℂ relative to the point a ∈ B. In terms of quadratic differentials, we give a complete description of extremal configurations in the problem of maximization of the functional ( r(B_1 ,a_1 ) r(B_3 ,a_3 ) | a_1 - a_3 |^2 )^α _1 ( r(B_2 ,a_2 ) r(B_4 ,a_4 ) | a_2 - a_4 |^2 )^α _2 defined on all collections consisting of points a_1, a_2, a_3, a_4 ∈ {z ∈ ℂ: |z| = 1} and pairwise-disjoint domains B_1, B_2, B_3, B_4 ⊂ ℂ such that a_1 ∈ B_1, a_1 ∈ B_2, a_3 ∈ B_3, and a_4 ∈ B_4., number=7, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Bakhtin, A. K., year=2006, month=Jul., pages=867–886