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 Trudy Inst. Mat. i Mekh. UrO RAN, 2020, Volume 26, Number 4, Pages 32–47 (Mi timm1764)

Analog of the Hadamard theorem and related extremal problems on the class of analytic functions

R. R. Akopyanab

a Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: We study several related extremal problems for analytic functions in a finitely connected domain $G$ with rectifiable Jordan boundary $\Gamma$. A sharp inequality is established between values of a function analytic in $G$ and weighted means of its boundary values on two measurable subsets $\gamma_1$ and $\gamma_0=\Gamma\setminus\gamma_1$ of the boundary:
$$|f(z_0)| \le \mathcal{C} \|f\|^{\alpha}_{L^{q}_{\varphi_1}(\gamma_1)} \|f\|^{\beta}_{L^{p}_{\varphi_0}(\gamma_0)},\quad z_0\in G, \quad 0<q, p\le\infty.$$
The inequality is an analog of Hadamard's three-circle theorem and the Nevanlinna brothers' theorem on two constants. In the case of a doubly connected domain $G$ and $1\le q,p\le\infty$, we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from a part of $\gamma_1$ to a given point of the domain. In these cases, the corresponding problems of optimal recovery of a function from its approximate boundary values on $\gamma_1$ and of the best approximation of a functional by linear bounded functionals are solved. The case of a simply connected domain $G$ has been completely investigated previously.

Keywords: analytic functions, optimal recovery of a functional, best approximation of an unbounded functional by bounded functionals, harmonic measure.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00336 Ural Federal University named after the First President of Russia B. N. Yeltsin 02.A03.21.0006 Ural Mathematical Center This work was supported by the Russian Foundation for Basic Research (project no.~18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University), and as part of research conducted in the Ural Mathematical Center.

DOI: https://doi.org/10.21538/0134-4889-2020-26-4-32-47

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Bibliographic databases:

UDC: 517.977
MSC: 30C85, 65E05, 30H99
Revised: 05.10.2020
Accepted:26.10.2020

Citation: R. R. Akopyan, “Analog of the Hadamard theorem and related extremal problems on the class of analytic functions”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 4, 2020, 32–47

Citation in format AMSBIB
\Bibitem{Ako20} \by R.~R.~Akopyan \paper Analog of the Hadamard theorem and related extremal problems on the class of analytic functions \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2020 \vol 26 \issue 4 \pages 32--47 \mathnet{http://mi.mathnet.ru/timm1764} \crossref{https://doi.org/10.21538/0134-4889-2020-26-4-32-47} \elib{https://elibrary.ru/item.asp?id=44314657}