I.V. Denega

• candidate of science
• Researcher at Institute of mathematics of NAS of Ukraine

In the paper, the open problem of the maximum product of the inner radii of $n$ domains in the case, when points and domains belong to the unit circle, is investigated. This problem is solved only for $n=2$ and $n=3$. No other results are known at present. We obtain the result for all $n \geqslant 2$. Also, we propose an approach that allows to est...

In this paper, estimates of products of the inner radii of non-overlapping domains containing points of some straight line have been obtained. The application of the obtained results to functions regular in a unit circle without common values has also been considered. Estimates of the derivatives of modules at more than two points for univalent fun...

In this paper, estimates of products of the inner radii of non-overlapping domains containing points of some straight line have been obtained. The application of the obtained results to functions regular in a unit circle without common values has also been considered. Estimates of the derivatives of modules at more than two points for univalent fun...

In 1934 Lavrentiev solved the problem of maximum ofproduct of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form$$T_{n}:={\prod\limits_{k=1}^nr(B_{k},a_{k})}{\bigg(\prod\limits_{1\leqslant k<p\leqslant n}|a_{k}-a_{p}|\...

In this paper, an approach which allowed to obtain new estimates of the products of the inner radii of mutually non-overlapping domains is proposed. Problem of the maximum of the product of inner radii of two non-overlapping multiconnected domains is considered.

In the paper we give a brief overview of the O. Bakhtin' scientific results

The result of M.A. Lavrentiev on the product of conformal radii of two non-overlapping simply connected domains has been generalized and strengthened. A method that allowed new estimates for the products of the inner radii of mutually non-overlapping domains to be obtained has been proposed.

The result of M.A. Lavrentiev on the product of conformal radii of two non-overlapping simply connected domains has been generalized and strengthened. A method that allowed new estimates for the products of the inner radii of mutually non-overlapping domains to be obtained has been proposed.

In the paper, we consider an open problem of finding the maximum of product of inner radii of mutually non-overlapping domains with respect to the points of the unit circle on a certain positive degree 𝛾 of the inner radius of the domain with respect to the origin, moreover, the domain containing origin does not intersect with other domains.

In the paper we obtain estimates of the maximums of products of generalized inner radii of mutually non-overlapping polycylindrical domains in \(\mathbb {C}^{n}\). The main theorems of the paper generalize and strengthening known results in the theory of non-overlapping domains with free poles on the unit circle onto the case of n-dimensional compl...

We consider the problem of maximization of the product of inner radii of n disjoint domains symmetric about the unit circle and of the 𝛾 th power of the inner radius of the domain with respect to zero. We solve the problem for n = 2, n = 3, and some 𝛾 > 1.

УДК 517.9Вивчається задача про максимум добутку внутрiшнiх радiусiв взаємно неперетинних областей, симетричних вiдносно одиничного кола, i внутрiшнього радiуса в додатному степенi деякої областi вiдносно початку координат.Розв’язано задачу про знаходження максимуму вказаного добутку при та деяких .

We establish effective upper estimates for the maximum products of the inner radii of mutually disjoint domains in (n,m)-radial systems of points of the complex plane for all possible values of a parameter γ. We also establish conditions under which the structure of points and domains is not important for our investigations.

Problems on extremal decomposition of the complex plane with free poles located on an (n;m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed.

Paper is devoted to extremal problems in geometric function theory of complex variables associated with estimates of functionals defined on systems of non-overlapping domains.

In this paper, the upper estimate for the maximum of the products of inner radii of mutually non-overlapping domains is obtained for any 𝑛-radial system of points on the complex plane at all possible values of some parameter 𝛾. The conditions under which the structure of points is not important in the proved results are established.

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. Based on these elementary estimates a number of new est...

Problems on extremal decomposition of the complex plane with free poles located on an (n,m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed.

We consider the problem of maximum of a functionalrγB00∐k=1nrBkak, where B0,…Bn n ≥ 2, are pairwise disjoint domains in C¯,a0 = 0, |ak| = 1,k=1,n,¯ and γ ϵ (0, n] (r(B, a) is the inner radius of the domain B⊂C¯ with respect to α). We show that this functional attains its maximum for a configuration of domains Bk and points ak with rotational n-symm...

Some extremal problems of the geometric theory of functions of a complex variable related to the estimates of functionals defined on systems of non-overlapping domains are considered. Till now, many such problems have not been solved, though some partial solutions are available. In the paper, the improved method is proposed for solving the problems...

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where \(B_{0}\), ..., \(B_{n}\), \(n\ge 2\), are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \(a_0 = 0\), \(|a_{k}| = 1\), \(k=\overline{1,n}\),...

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the...

Some extremal problems of the geometric theory of functions of a complex variable related to the estimates of functionals defined on systems of non-overlapping domains are considered. Till now, many such problems have not been solved, though some partial solutions are available. In the paper, the improved method is proposed for solving the problems...

We study the following problem: Let a0 = 0, ∣a1 ∣ = … = ∣ an ∣ = 1, \( {a}_k\in {B}_k\subset \overline{\mathbb{C}} \), where B0, … , Bn are mutually disjoint domains and B1, … , Bn are symmetric about the unit circle. It is necessary to find the exact upper bound for the product \( {r}^{\gamma}\left({B}_0,0\right){\prod}_{k=1}^nr\left({B}_k,{a}_k\r...

In geometric function theory of a complex variable problems on extremal decomposition with free poles on the unit circle are well known. One of such problem is the problem on maximum of the functional $${r^\gamma }({B_0},0)\prod\limits_{k = 1}^n r ({B_k},{a_k}),$$ where B0 , B1 , B2 ,..., Bn , n ≥ 2, are pairwise disjoint domains in ¯𝔺, a0 = 0, | a...

In the paper we study a generalization of the extremal problem of geometric theory of functions of a complex variable on non-overlapping domains with free poles: Fix any γ ∈ R + and find the maximum (and describe all extremals) of the functional [r (B 0 , 0) r (B ∞ , ∞)] γ Π n k=1 r (B k , a k ) , where n ∈ N, n >= 2, a 0 = 0, |a k | = 1, B 0 , B ∞...

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. In this paper, we consider the well-known problem of ma...

where n ∈ ℕ, n ≥ 2, γ ∈ ℝ⁺, \( {A}_n={\left\{{a}_k\right\}}_{k=1}^n \) is a system of points such that |ak| = 1, a0 = 0, B0, B∞, \( {\left\{{B}_k\right\}}_{k=1}^n \) is a system of pairwise nonoverlapping domains, \( {a}_k\in {B}_k\subset \overline{\mathrm{\mathbb{C}}} \), \( k=\overline{0,n} \), \( \infty \in {B}_{\infty}\subset \overline{\mathrm{...

The paper deals with the following open problem stated by V.N. Dubinin. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\subset \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains. For all values of the parameter $\gamma\in (0, n]$ find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where...

We propose a generalization of the notion of the inner radius to the case of n-dimensional complex space, which allows us to transfer some results known for the complex plane onto \( {\mathrm{\mathbb{C}}}^n \).

The paper is devoted to investigation of the problems of geometric function theory of a complex variable.

Paper is devoted to one classic problem of geometric function theory on extremal decomposition of the complex plane. We consider a problem of maximization of product of inner radii of n non-overlapping domains.

This paper is devoted to one open extremal problem of Dubinin in geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider Dubinin’s problem of the maximum of product of inner radii of n non-overlapping domains containing points of the unit circle and the...

Within the geometric theory of functions, we study one of the classical problems of extreme decomposition of a complex plane.

The paper is devoted to investigation of the problems of geometric function theory of a complex variable. A general problem of the description of extremal configurations maximizing the product of the inner radii of mutually non-overlapping domains is studied.

The report is devoted to investigation of the problems of geometric function theory of a complex variable.

Considered here is one quite general problem about description of extremal configurations maximizing the product of inner radii non-overlapping domains.

Some results related to extremal problems with free poles on radial systems are generalized. They are obtained by applying the known methods of geometric function theory of complex variable. Sufficiently good numerical results for are obtained.

In [1] was proposed a variant of the construction of a commu- tative an associative algebra in Cn. We denote by Σm a set of systems of pairs (Bk,Ωk) of polycylindrical domains and points considered in theorem 3 [3]. A following theorem is proved (all definitions see in [2, 3]).

We study one extremal problem on the product of power of generalized inner radii of non-overlapping domains in $\mathbb{C}^{n}$.

The paper is devoted to extremal problems of the geometric function theory of complex variable associated with estimates of functionals defined on systems of non-overlapping domains. In particular, the investigation is focused on the strengthening and generalization of some known results in this theory. 1. Introduction. In geometric function theory...

In this paper we consider quite general problem on non-overlapping domains with free poles on radial systems. The main theorem of this work generalizes the previously known results for problems of this type.

Although much research has been devoted to extremal problems on non-overlapping domains little is known about all solutions of this problems. We generalized some of this problems on the case of more general systems of points. It was solved using separating transformations and learning functions in detail. Methods used in the paper allowed to get im...

Paper is devoted to extremal problems of geometric function theory with estimates of functionals defined on systems of non-overlapping domains. In particular, focus of investigation is well-known problem of V.N. Dubinin and generalization of some results in this problem.

Extremal Problems of Functions Complex Variable In geometric theory of functions of complex variable extremum problems of non-overlapping domains are well known classic direction. The fundamental work [1] has served the initial impetus for the emergence of such direction, in which, in particular, was first recognized and solved the problem of maxim...