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 (Formula presented.) of the boundary: (Formula presented.) The inequality is an analog of Hadamard’s three-circle theorem and the Nevanlinna brothers’ two-constant theorem.In the case of a doubly connected domain G and (Formula presented.), 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 the part (Formula presented.) of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on (Formula presented.) and the problem of the best approximation of a functional by bounded linear functionals are solved.The case of a simply connected domain G has been completely investigated previously. © 2021, Pleiades Publishing, Ltd.