Considered here is one quite general problem about the description of extremal configurations maximizing the product of inner radii mutually non-overlapping domains the next following form: \begin{equation} J_{n}(\gamma)=\left[r\left(B_0,0\right)r\left(B_\infty,\infty\right)\right]^{\gamma}\prod\limits_{k=1}^n r\left(B_k,a_k\right),\end{equation} where $\gamma\in\mathbb{R^{+}}$, $A_{n}=\{a_{k}\}_{k=1}^{n}$ -- $n$-radial points system, $B_0$, $B_\infty$, $\{B_{k}\}_{k=1}^{n}$ -- set of systems of mutually disjoint domains, \textup{(}$a_{0}=0\in B_0\subset{\mathbb{C}}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset{\mathbb{C}}$\textup{)}, achieved for some configuration of domains $B_{0}$, $B_{k}$, $B_{\infty}$ and points $a_{0}$, $a_{k}$, $\infty$, $k=\overline{1, n}$. The functional (1) evaluation for the first time was obtained in 1988 by V.M. Dubinin \cite{4} for {$\gamma=\frac{1}{2}$ и $n\geqslant2$} for systems of disjoint domains using symmetrization method. A special case, when domains are univalent, was examined by G.V. Kuzmina in \cite{KYZMINA-01} After the result of V.M. Dubinin in the general formulation for the arbitrary multiply-concted domains was not until 2017. In 2017 in the work of I. Dengi, A. Targonsky \cite{iratarg} was the result for $\gamma_{n}=0.08n^{2}$, $n\geq7$. The result was obtained through a lower bound of $min_t (n-1)x_{1}+x_{2},$ where $x_{1}(t)+x_{2}(t)$ is the equation $F^\prime (x)=t$ solution, $F^\prime (x)=4x\ln(x)-2(x-1)\ln|x-1|-2(x+1)\ln(x+1)+\frac{2}{x}, $ $y_0 \leqslant t<0$, $y_{0}\approx-1,0599$. In this paper, a much better estimate of $min_t (n-1)x_{1}+x_{2}$ was obtained through a lower bound with the specified parameters. On the basis of this, the article succeeded in obtaining an estimate of the maximum of the functional (1) over a larger interval of the parameter $\gamma$, $\gamma\in(0,\gamma_{n}]$, $n\geq7$. Received the result for for any points of the circle $|a_k|=R$, $k=\overline{1,n}$, and any pairwise disjoint systems of domains $B_k$, $a_{0}=0\in B_0\subset{\mathbb{C}}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset{\mathbb{C}}$, $k=\overline{1,n}.$