The problem of the product of inner radii of four nonoverlapping domains, some of there are symmetric about unit circle

Considered in the paper is one quite general problem of geometric function theory on extremal decomposition of the complex plane, namely to determine the maximum of product of the inner radii of $n$ non-overlapping domains $\{B_{k}\}_{k=1}^{n},$, symmetric with respect to the unit circle, and the power $\gamma$ of the inner radius of a domain $\{B_ {0}\}$, which contains the origin. Starting point of the theory of extremal problems on non-overlapping domains is the result of Lavrent’ev \cite{Lavr} who in 1934 solved the problem of a product of conformal radii of two mutually nonoverlapping simply connected domains. It was the first result of this direction. Goluzin \cite{Goluzin} generalized this problem in the case of an arbitrary finite number of mutually disjoint domains and obtained an accurate evaluation for the case of three domains. Further, Kuzmina \cite{Kuzm} showed that the problem of the evaluation for the case of four domains is reduced to the smallest capacity problems in a certain continuum family and received the exact inequality for $n=4$. For $n\geq5$ full solution of the problem is not obtained at this time. The problem, considered in this paper, stated in \cite{Dubinin-1994} by V.N. Dubinin and earlier in different form by G.P. Bakhtina \cite{Bakhtina-1984}. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\in \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains, and $B_{1},\ldots, B_{n}$ are symmetric about the unit circle. Find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where $r(B_k,a_k)$ is the inner radius of $B_k$ with respect to $a_k$. For $\gamma=1$ and $n\geq2$ this problem was solved by L.V. Kovalev \cite{kovalev-2000,kovalev2-2000} and for $\gamma_{n}=0,38n^{2}$ and $n\geq2$ under the additional assumption that the maximum $\alpha_{0}$ of the angles between neighbouring line segments $[0, a_{k}]$ do not exceed $2\pi/\sqrt{2\gamma}$ it was solved in \cite{BahDenV}. In the present paper this problem is solved for three non-overlapping symmetric domains and for $0<\gamma\leq1.233$ without additional restrictions, moreover, for the first time such $1<\gamma$ are considered for this case. Was proved the lemma, by which it was obtained the estimate of the inner radius of a domain $\{B_ {0}\}$, which contains the origin. Using this lemma and the result of paper \cite{BahDenV}, it was proved that for $\alpha_{0}>2\pi/\sqrt{2\gamma}$ consided product does not exceed some expression.