# Bakhtin A. K.

### Inequalities for the inner radii of nonorevlapping domains

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 7. - pp. 996-1002

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 $\gamma\in (1, n]$.

### Inequalities for inner radii of symmetric disjoint domains

Bakhtin A. K., Denega I. V., Vyhovs'ka L.V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 9. - pp. 1282-1288

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 $[0, a_k]$.

### Estimates of the product of inner radii of five nonoverlapping domains

Bakhtin A. K., Dvorak I. Ya., Zabolotnyi Ya. V.

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 261-267

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.

### Yurii Ivanovych Samoilenko (on the 80th anniversary of his birthday)

Bakhtin A. K., Gerasimenko V. I., Plaksa S. A., Samoilenko A. M., Sharko V. V., Trohimchuk Yu. Yu, Yacenko V. O., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 574-576

### Generalized (*n, d*)-ray systems of points and inequalities for nonoverlapping domains and open sets

Bakhtin A. K., Targonskii A. L.

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 867-879

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

### Inequalities for the inner radii of nonoverlapping domains and open sets

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 596-610

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

### Sharp estimates for inner radii of systems of nonoverlapping domains and open sets

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1601–1618

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.

### Application of a separating transformation to estimates of inner radii of open sets

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1313–1321

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.

### Some extremal problems in the theory of nonoverlapping domains with free poles on rays

Bakhtin A. K., Targonskii A. L.

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1715–1719

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.

### Extremal problems of nonoverlapping domains with free poles on a circle

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 867–886

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 $a_1 ∈ B_1, a_1 ∈ B_2, a_3 ∈ B_3, and a_4 ∈ B_4$.

### Some problems in the theory of nonoverlapping domains

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 723–731

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

### On the product of inner radii of symmetric nonoverlapping domains

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1454–1464

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.

### On extremal problems for symmetric disjoint domains

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 179–185

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

### Coefficients of univalent functions of the Gel'fer class

Ukr. Mat. Zh. - 1985. - 37, № 6. - pp. 683–689

### Some properties of functions of class S

Ukr. Mat. Zh. - 1981. - 33, № 2. - pp. 154–159

### On some extremal problems in conformal mapping

Ukr. Mat. Zh. - 1974. - 26, № 4. - pp. 517–522