Estimates of the inner radii of non-overlapping domains

We consider the well-known problem of the geometric theory of functions of a complex variable on non-overlapping domains with free poles on radial systems. The main results of the present work strengthen and generalize several known results for this problem.

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Some estimates for extremal decomposition of the complex plane

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2018-32-5 ◽

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 $\overline{\mathbb{C}}$.

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Extremal problems on the generalized (n, d)-equiangular system of points

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2014-0044 ◽

2014 ◽

Vol 22 (2) ◽

pp. 239-252 ◽

Cited By ~ 4

Author(s):

A. Targonskii

Keyword(s):

Complex Variable ◽

Extremal Problems ◽

Geometrical Theory ◽

Overlapping Domains

AbstractThe paper of Lavrent’ev [1] was the beginning of geometrical theory of functions of the complex variable. He solved a problem on the product of conformal radiuses of two non-overlapping domains. In many papers (see [2] – [13]) the Lavrent’ev’s result are generalized. In this paper are obtained the new results of this direction.

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Estimation of the products of the inner radii of domains with an additional symmetry condition

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2019-33-6 ◽

2019 ◽

Vol 33 ◽

pp. 83-90

Author(s):

Iryna Denega

Keyword(s):

Conformal Mapping ◽

Complex Variable ◽

Zero Point ◽

Convergent Series ◽

Extremal Problems ◽

Symmetry Condition ◽

Additional Symmetry ◽

Geometric Function ◽

Overlapping Domains ◽

Upper Estimate

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 estimates for functions realizing a conformal mapping of a disc onto domains with certain special properties are obtained. Estimates of this type are fundamental to solving some metric problems arising when considering the cor\-res\-pon\-dence of boundaries under a conformal mapping. Also, on the basis of the results concerning various extremal properties of conformal mappings, the problem of the representability of functions of a complex variable by a uniformly convergent series of polynomials is solved. In this paper, we consider the problem on maximum the products of the inner radii of $n$ disjoint domains with an additional symmetry condition that contain points of extended complex plane and the degree $\gamma$ of the inner radius of the domain that contains the zero point. An upper estimate for the maximum of this product is found for all values of $\gamma\in(0,\,n]$. The main result of the paper generalizes and strengthens the results of the predecessors [1-4] for the case of an arbitrary arrangement of points systems on $\overline{\mathbb{C}}$. In proving the main theorem, the arguments of proving of Lemma 1 [5] and the ideas of proving Theorem 1 [3] played a key role. We also established the conditions under which the structure of points and domains is not important. The corresponding results are obtained for the case when the points are placed on the unit circle and in the case of any fixed $n$-radial system of points.

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Extremal decomposition of the complex plane with free poles. II

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2019-16-4-2 ◽

2019 ◽

Vol 16 (4) ◽

pp. 477-495

Author(s):

Aleksandr Bakhtin ◽

Iryna Denega

Keyword(s):

Complex Plane ◽

Upper Bounds ◽

Extremal Decomposition ◽

Overlapping Domains

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.

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A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-2-7 ◽

2020 ◽

Vol 17 (2) ◽

pp. 256-277

Author(s):

Ol'ga Veselovska ◽

Veronika Dostoina

Keyword(s):

Complex Plane ◽

Complex Variable ◽

Analytic Functions ◽

Combinatorial Identities ◽

Independent Interest ◽

Closed Curves ◽

In Series ◽

Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

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The Uniformisation of the Equation $$z^w=w^z$$

Computational Methods and Function Theory ◽

10.1007/s40315-021-00369-6 ◽

2021 ◽

Author(s):

A. F. Beardon

Keyword(s):

Positive Solutions ◽

Complex Plane ◽

Complex Variable ◽

Holomorphic Functions ◽

Parametric Form ◽

Lambert Function ◽

The Real ◽

Complex Equation ◽

Real Equation ◽

Expository Paper

AbstractThe positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x .

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