scholarly journals About one extremal problem for open sets and partially non-overlapping domains

2019 ◽  
Vol 16 (2) ◽  
pp. 228-238
Author(s):  
Andrey Targonskii ◽  
Irina Targonskaya ◽  
Kateryna Vaschenko
Keyword(s):  
Finite Number ◽  
Extremal Problem ◽  
Pairwise Disjoint ◽  
Sharp Estimates ◽  
Overlapping Domains ◽  
Disjoint Domains

Sharp estimates of a product of the inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of any finite number of free poles on the rays.

scholarly journals On the One Extremal Problem on the Riemann Sphere

2016 ◽  
Vol 4 ◽  
pp. 1-7 ◽  
Author(s):  
Andrey L. Targonskii ◽  
Irina Targonskaya
Keyword(s):  
Finite Number ◽  
Extremal Problem ◽  
Pairwise Disjoint ◽  
Riemann Sphere ◽  
Sharp Estimates ◽  
The One ◽  
Disjoint Domains

Sharp estimates of product of inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of arbitrary finite number of rays containing arbitrary even number of free poles.

scholarly journals On the One Extremal Problem with the Free Poles on the Unit Circle

2016 ◽  
Vol 6 ◽  
pp. 26-31 ◽  
Author(s):  
Andrey L. Targonskii
Keyword(s):  
Finite Number ◽  
Unit Circle ◽  
Extremal Problem ◽  
Pairwise Disjoint ◽  
Sharp Estimates ◽  
Inner Radius ◽  
The One ◽  
Disjoint Domains

The sharp estimates of the product of the inner radius for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of an arbitrary finite number of the free poles on the unit circle for the following functional (see formula in paper)

Extremal decomposition of the complex plane with free poles

2019 ◽  
Vol 16 (3) ◽  
pp. 307-328
Author(s):  
Aleksandr Bakhtin ◽  
Iryna Denega
Keyword(s):  
Extremal Problem ◽  
Pairwise Disjoint ◽  
Additional Assumption ◽  
Function Theory ◽  
Complex Variables ◽  
Geometric Function Theory ◽  
Line Segments ◽  
Geometric Function ◽  
Inner Radius ◽  
Disjoint Domains

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 \(\overline{\mathbb{C}}\), \(a_0 = 0\), \(|a_{k}| = 1\), \(k=\overline{1,n}\), and \(\gamma\in (0, n]\) (\(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to a point \(a\in B\)). For all values of the parameter \(\gamma\in (0, n]\), it is necessary to show that its maximum is attained for a configuration of domains \(B_{k}\) and points \(a_{k}\), \(k=\overline{0,n}\), possessing the \(n\)-fold symmetry. The problem was solved by V.N. Dubinin [1, 2] for \(\gamma=1\) and by G.V. Kuz’mina [4] for \(0 \lt \gamma \lt 1\). L.V. Kovalev [4] obtained its solution for \(n \ge 5\) under the additional assumption that the angles between neighbouring line segments \([0, a_{k}]\) do not exceed \(2\pi /\sqrt{\gamma}\). In particular, this problem will be solved in the present paper for \(n=2\) and \(\gamma\in(1,\,2]\).

scholarly journals The problem of extreme decomposition of the plane

2018 ◽  
Vol 32 ◽  
pp. 3-9
Author(s):  
Aleksandr Bakhtin ◽  
Inna Dvorak
Keyword(s):  
Lower Bound ◽  
General Problem ◽  
Pairwise Disjoint ◽  
Symmetrization Method ◽  
Overlapping Domains ◽  
First Time ◽  
Special Case ◽  
Disjoint Domains

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 $B_k$, $a_{0}=0\in B_0\subset{\mathbb{C}}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset{\mathbb{C}}$, $k=\overline{1,n}.$

scholarly journals Some estimates for extremal decomposition of the complex plane

2018 ◽  
Vol 32 ◽  
pp. 42-47
Author(s):  
Iryna Denega
Keyword(s):  
Complex Variable ◽  
Open Problem ◽  
Function Theory ◽  
Extremal Problems ◽  
Geometric Function Theory ◽  
Geometric Function ◽  
Inner Radius ◽  
Overlapping Domains ◽  
Disjoint Domains

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 maximum of the functional \(r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right)\), where \(B_{0}\),..., \(B_{n}\) are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \( a_0=0 \), \(|a_{k}|=1\), \(k=\overline{1,n}\) are different points of the circle, \(\gamma\in (0, n]\), and \(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to the point \( a \). This problem was posed as an open problem in the Dubinin paper in 1994. Till now, this problem has not been solved, though some partial solutions are available. In the paper an estimate for the inner radius of the domain that contains the point zero is found. The main result of the paper generalizes the analogous results of [1, 2] to the case of an arbitrary arrangement of systems of points on \(\overline{\mathbb{C}}\).

scholarly journals On the Closure of the Image of the Generalized Divisor Function

2017 ◽  
Vol 12 (2) ◽  
pp. 77-90 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Finite Number ◽  
Real Number ◽  
Closed Form ◽  
Pairwise Disjoint ◽  
Arithmetic Function ◽  
Divisor Function ◽  
Closed Intervals ◽  
Topological Closure ◽  
Positive Integers ◽  
Sufficient Precision

Abstract For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , Iℓ. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , Iℓ and d1, . . . , dℓ, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.

The problem of the product of inner radii of four nonoverlapping domains, some of there are symmetric about unit circle

2019 ◽  
Vol 33 ◽  
pp. 27-32
Author(s):  
Alexandr Bakhtin ◽  
Yaroslav Zabolotnii
Keyword(s):  
Unit Circle ◽  
Extremal Problems ◽  
Exact Inequality ◽  
Geometric Function ◽  
Starting Point ◽  
Inner Radius ◽  
Overlapping Domains ◽  
Full Solution ◽  
Simply Connected ◽  
Disjoint Domains

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.

Tarski and geometry

1986 ◽  
Vol 51 (4) ◽  
pp. 907-912 ◽  
Author(s):  
L. W. Szczerba
Keyword(s):  
Finite Number ◽  
Pairwise Disjoint ◽  
Axiom System ◽  
Proper Subset

Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.

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