Analog of the Hadamard Theorem and Related Extremal Problems on the Class of Analytic Functions

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.