Extremal decomposition of the complex plane with free poles. II

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.

Problem on extremal decomposition of the complex plane with free poles

2020 ◽  
Vol 17 (1) ◽  
pp. 3-29
Author(s):  
Aleksandr Bakhtin ◽  
Liudmyla Vyhivska
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Geometric Theory ◽  
Extremal Decomposition ◽  
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.

scholarly journals Estimates of the inner radii of non-overlapping domains

2019 ◽  
Vol 16 (1) ◽  
pp. 46-56
Author(s):  
Iryna Denega
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Geometric Theory ◽  
Extremal Problems ◽  
Improved Method ◽  
Extremal Decomposition ◽  
Overlapping Domains

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 on extremal decomposition of the complex plane. The main results generalize and strengthen some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane.

Extremal decomposition of the complex plane with free poles

2020 ◽  
Vol 246 (1) ◽  
pp. 1-17
Author(s):  
Aleksandr K. Bakhtin ◽  
Iryna V. Denega
Keyword(s):  
Complex Plane ◽  
Extremal Decomposition

scholarly journals Approximation of classes of Poisson integrals by Fejer sums

2018 ◽  
Keyword(s):  
Complex Plane ◽  
Upper Bounds ◽  
Periodic Functions ◽  
Poisson Integrals

For upper bounds of the deviations of Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov-Nikolsky problem.

On some higher order equations admitting meromorphic solutions in a given domain

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Grigor Barsegian ◽  
Fanning Meng
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Recent Trend ◽  
Upper Bounds ◽  
Higher Order ◽  
General Nature ◽  
Meromorphic Solutions ◽  
Complex Differential Equations ◽  
Similar Solutions ◽  
Higher Order Differential Equations

AbstractThis paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a-points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions.

scholarly journals Weakened problem on extremal decomposition of the complex plane

2019 ◽  
Vol 51 (1) ◽  
Author(s):  
A. K. Bakhtin ◽  
I. V. Denega ◽  
Yu. V. Shunkin
Keyword(s):  
Complex Plane ◽  
Extremal Decomposition
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