
Analog of the Hadamard theorem and related extremal problems on the class of analytic functions
R. R. Akopyan^{ab} ^{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 threecircle 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.
DOI:
https://doi.org/10.21538/0134488920202643247
Full text:
PDF file (283 kB)
References:
PDF file
HTML file
Bibliographic databases:
UDC:
517.977
MSC: 30C85, 65E05, 30H99 Received: 13.07.2020 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 3247
\mathnet{http://mi.mathnet.ru/timm1764}
\crossref{https://doi.org/10.21538/0134488920202643247}
\elib{https://elibrary.ru/item.asp?id=44314657}
Linking options:
http://mi.mathnet.ru/eng/timm1764 http://mi.mathnet.ru/eng/timm/v26/i4/p32
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  39  Full text:  4  References:  1 
