Extremal decomposition of the complex plane with free poles

Abstract

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where \(B_{0}\), ..., \(B_{n}\), \(n\ge 2\), are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \(a_0 = 0\), \(|a_{k}| = 1\), \(k=\overline{1,n}\), and \(\gamma\in (0, n]\) (\(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to a point \(a\in B\)). For all values of the parameter \(\gamma\in (0, n]\), it is necessary to show that its maximum is attained for a configuration of domains \(B_{k}\) and points \(a_{k}\), \(k=\overline{0,n}\), possessing the \(n\)-fold symmetry. The problem was solved by V.N. Dubinin [1, 2] for \(\gamma=1\) and by G.V. Kuz’mina [4] for \(0 \lt \gamma \lt 1\). L.V. Kovalev [4] obtained its solution for \(n \ge 5\) under the additional assumption that the angles between neighbouring line segments \([0, a_{k}]\) do not exceed \(2\pi /\sqrt{\gamma}\). In particular, this problem will be solved in the present paper for \(n=2\) and \(\gamma\in(1,\,2]\).