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 \(\Gamma\). 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: \(|f(z_{0})|\leq\mathcal{C}\,\|f\|^{\alpha}_{L^{q}_{\varphi_{1}}(\gamma_{1})}\, \|f\|^{\beta}_{L^{p}_{\varphi_{0}}(\gamma_{0})},z_{0}\in G,0<q,p\leq\infty.\) 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 \(1\leq q,p\leq\infty\), 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 the part \(\gamma_{1}\) 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 \(\gamma_{1}\) 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.