Please use this identifier to cite or link to this item: http://hdl.handle.net/10995/111244
Title: Analog of the Hadamard Theorem and Related Extremal Problems on the Class of Analytic Functions
Other Titles: Аналог теоремы Адамара и связанные экстремальные задачи на классе аналитических функций
Authors: Akopyan, R. R.
Issue Date: 2020
Publisher: Krasovskii Institute of Mathematics and Mechanics
Krasovskii Institute of Mathematics and Mechanics UB RAS
Citation: Akopyan R. R. Analog of the Hadamard Theorem and Related Extremal Problems on the Class of Analytic Functions [Аналог теоремы Адамара и связанные экстремальные задачи на классе аналитических функций] / R. R. Akopyan // Trudy Instituta Matematiki i Mekhaniki UrO RAN. — 2020. — Vol. 26. — Iss. 4. — P. 32-47.
Abstract: We study several related extremal problems for analytic functions in a finitely connected domain G with rectifiable Jordan boundary Γ. A sharp inequality is established between values of a function analytic in G and weighted means of its boundary values on two measurable subsets γ1 and γ0 = Γ \ γ1 of the boundary: |f(z0)| ≤ C kfkαLqϕ1 (γ1) kfkβLpϕ0 (γ0), z0 ∈ G, 0 < q, p ≤ ∞. 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 ≤ q, p ≤ ∞, 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 γ1 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 γ1 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. © 2020 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.
Keywords: ANALYTIC FUNCTIONS
BEST APPROXIMATION OF AN UNBOUNDED FUNCTIONAL BY BOUNDED FUNCTIONALS
HARMONIC MEASURE
OPTIMAL RECOVERY OF A FUNCTIONAL
URI: http://hdl.handle.net/10995/111244
Access: info:eu-repo/semantics/openAccess
SCOPUS ID: 85103642873
PURE ID: 20232586
ISSN: 0134-4889
metadata.dc.description.sponsorship: 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.
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