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.
On the One Extremal Problem with the Free Poles on the Unit Circle
