@imath.kiev.ua
Complex analysis and potential theory department
Institute of mathematics of NAS of Ukraine
Analysis, Algebra and Number Theory
Scopus Publications
Yaroslav Zabolotnyi and Iryna Denega
Walter de Gruyter GmbH
Abstract In the paper, an approach is proposed that allowed to establish new upper estimates for products of inner radii of mutually non-overlapping domains.
Iryna Denega and Yaroslav Zabolotnyi
Springer Science and Business Media LLC
Iryna Denega and Yaroslav Zabolotnyi
Springer Nature Switzerland
Yaroslav Zabolotnyi and Iryna Denega
Informa UK Limited
Iryna Denega and Yaroslav Zabolotnyi
Odesa National University of Technology
In the paper an extreme problem of geometric function theory of a complex variable on the maximum of product of the inner radii on a system of n mutually non-overlapping multiply connected domains Bk containing the fixed points ak, k=1,...,n, located on a hyperbola branches is investigated.
Iryna Denega and Yaroslav Zabolotnyi
Springer Science and Business Media LLC
Iryna Denega and Yaroslav Zabolotnyi
Springer Science and Business Media LLC
I. Denega and Ya. V. Zabolotnyi
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 establish evolutionary inequalities for the products of the inner radii of mutually non-overlapping domains.
Iryna Denega
Springer International Publishing
Iryna Denega and Yaroslav Zabolotnyi
Odesa National University of Technology
In the paper we give a brief overview of the O. Bakhtin' scientific results
I. V. Denega and Ya. V. Zabolotnyi
Ivan Franko National University of Lviv
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$$T_{n}:={\\prod\\limits_{k=1}^nr(B_{k},a_{k})}{\\bigg(\\prod\\limits_{1\\leqslant k<p\\leqslant n}|a_{k}-a_{p}|\\bigg)^{-\\frac{2}{n-1}}},$$where $r(B,a)$ denotes the inner radius of the domain $B$ with respect to the point $a$ (for an infinitely distant point under the corresponding factor we understand the unit).In 1951 Goluzin for $n=3$ obtained an accurate evaluation for $T_{3}$.In 1980 Kuzmina showedthat the problem of the evaluation of $T_{4}$ isreduced to the smallest capacity problem in the certain continuumfamily and obtained the exact inequality for $T_{4}$.No other ultimate results in this problem for $n \\geqslant 5$ are known at present.In 2021 \\cite{Bakhtin2021,BahDen22} effective upper estimates are obtained for $T_{n}$, $n \\geqslant 2$.Among the possible applications of the obtained results in other tasks of the function theory are the so-called distortion theorems.In the paper we consider an application of upper estimates for products of inner radii to distortion theorems for univalent functionsin disk $U$, which map it onto a star-shaped domains relative to the origin.
Aleksandr K. Bakhtin and Iryna V. Denega
Springer Science and Business Media LLC
Iryna Denega
Springer International Publishing
A. K. Bakhtin, L. V. Vyhivska, and I. V. Denega
Springer Science and Business Media LLC
Ya. Zabolotnii and I. Denega
Petrozavodsk State University
Yaroslav Zabolotnii, , Iryna Denega, and
L. N. Gumilyov Eurasian National University
A. K. Bakhtin and I. V. Denega
Springer Science and Business Media LLC
Aleksandr K. Bakhtin and Iryna V. Denega
Springer Science and Business Media LLC
Aleksandr K. Bakhtin and Iryna V. Denega
Springer Science and Business Media LLC
I. Denega
Petrozavodsk State University
A. K. Bakhtin and I. V. Denega
Springer Science and Business Media LLC
Iryna Denega
Springer Science and Business Media LLC
Yaroslav Zabolotnii and Iryna Denega
Springer Science and Business Media LLC
A. K. Bakhtin, I. V. Denega, and L. V. Vygovskaya
Springer Science and Business Media LLC