Bakhtin A. K.

UDC 517.54
We study a problem of nonoverlapping domains with free poles on radial systems. Our main results strengthen and generalize several known results obtained in the investigation of this problem.

UDC 517.54
We consider the problem of maximum of the functional $$ r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right), $$ where $B_{0},\ldots,B_{n}$, $n\geq 2$, 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 $B\subset\overline{\mathbb{C}}$ with respect to $a$). Show that it attains its maximum at a configuration of domains $B_{k}$ and points $a_{k}$ possessing rotational $n$-symmetry. This problem was solved by Dubinin for $\gamma=1$ and by Kuz’mina for $0<\gamma< 1$. Later, Kovalev solved this problem for $n\geqslant5$ under an additional assumption that the angles between neighboring linear segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$. We generalize this problem to the case of arbitrary locations of the systems of points in the complex plane and obtain some estimates for the functional for all $n$ and $\gamma\in (1, n]$.

We study the following problem: Let $a_0 = 0, | a_1| = ... = | a_n| = 1,\; a_k \in B_k {\subset C}$, where $B_0, ... ,B_n$ are disjoint domains, and $B_1, ... ,B_n$ are symmetric about the unit circle. It is necessary to find the exact upper bound for $r^{\gamma} (B_0, 0) \prod^n_{k=1} r(B_k, a_k)$, where $r(B_k, a_k)$ is the inner radius of Bk with respect to $a_k$. For $\gamma = 1$ and $n \geq 2$, the problem was solved by L. V. Kovalev. We solve this problem for $\gamma \in (0, \gamma_n], \gamma_n = 0,38 n^2$, and $n \geq 2$ under the additional assumption imposed on the angles between the neighboring line segments $[0, a_k]$.

We study the extremal V. N. Dubinin problem in the geometric theory of functions of complex variables connected with the estimates of a functional defined on a system of nonoverlapping domains. A particular solution of this problem is obtained.

We solve the extremal problem of finding the maximum of the functional.

We generalize some classical results in the theory of extreme problems for nonoverlapping domains.

We study extremal problems of the geometric theory of functions of a complex variable. Sharp upper estimates are obtained for the product of inner radii of disjoint domains and open sets with respect to equiradial systems of points.

We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets.

We obtain new results on the maximization of the product of powers of the interior radii of pairwise disjoint domains with respect to certain systems of points in the extended complex plane.

Let $α_1, α_2 > 0$ and let $r(B, a)$ be the interior radius of the domain $B$ lying in the extended complex plane $\overline{ℂ}$ relative to the point $a ∈ B$. In terms of quadratic differentials, we give a complete description of extremal configurations in the problem of maximization of the functional $\left( {\frac{{r(B_1 ,a_1 ) r(B_3 ,a_3 )}}{{\left| {a_1 - a_3 } \right|^2 }}} \right)^{\alpha _1 } \left( {\frac{{r(B_2 ,a_2 ) r(B_4 ,a_4 )}}{{\left| {a_2 - a_4 } \right|^2 }}} \right)^{\alpha _2 }$ defined on all collections consisting of points $a_1, a_2, a_3, a_4 ∈ \{z ∈ ℂ: |z| = 1\}$ and pairwise-disjoint domains $B_1, B_2, B_3, B_4 ⊂ \overline{ℂ}$ such that $a_1 ∈ B_1, a_1 ∈ B_2, a_3 ∈ B_3, and a_4 ∈ B_4$.

We generalize some results concerning extremal problems of nonoverlapping domains with free poles on the unit circle.

Some results concerning extremal problems for nonoverlapping domains with free poles on the unit circle, known for the simply connected case, are generalized to the case of multiply connected domains.

We study two extremal problems for the product of powers of conformal radii of symmetric disjoint domains.

We study an analog of the problem on then th diameters of continua in the complex plane.