The problem of the maximum of product of inner radii of non-overlapping domains, some of there are symmetric with respect to the 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. The start point of the theory of extremal problems on non-overlapping domains was the Lavrent’ev theorem, who in 1934 solved the problem of the maximum of product of conformal radii of two simply connected non-overlapping domains. It was the first result of this direction. For three fixed poles and three non-overlapping domains the similar problem was solved by Goluzin in 1951, for four points and four domains -- by Kuzmina in 1980. For $n\geq5$ full solution of the problem is not obtained at this time. The problems of extremal decomposition of the complex plane were studied in many papers, but, despite this, many extremal problems are currently unsolved. 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-2000a,Kovalev-2000b} 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}. Also, in the paper \cite{BahDen-2019}, this problem was solved for the case of an arbitrary natural $n\geq8$ and $1<\gamma<\sqrt{n}$, and in paper \cite{Zabolotnyi-2020} -- for $n\geq 20$ and $1<\gamma\leq n^{\frac{2}{3}-q(n)}$, where $q(n)=\frac{2}{3}\frac{\ln\left(2\sqrt{2}\ln(n)\right)}{\ln(n)}$. In the present paper this problem is solved for all $n\geq258$ non-overlapping symmetric domains and for $0<\gamma\leq n$ with with an additional condition that $r(B_0,0)\leq1$, moreover, for $0<\gamma\leq n^{\alpha}$ for $0<\alpha<1$ it was shown that the number $n$ can be significantlythe reduce. Using the result of paper \cite{BahDen-2019}, this problem was solved for all $n\geq44$ and $0<\gamma\leq n^{\alpha}$ for $0<\alpha<\frac{2}{3}-\frac{2}{3}\frac{\ln\left(2\ln(n)\right)}{\ln(n)}$ without the condition that $r(B_0,0)\leq1$