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

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 function and weighted means its on two measurable subsets $$\gamma_{1}$$ $$\gamma_{0}=\Gamma\setminus\gamma_{1}$$ 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 an analog Hadamard’s three-circle theorem Nevanlinna brothers’ two-constant theorem. In case doubly $$1\leq q,p\leq\infty$$ , we cases where provides value modulus continuity functional extension from part to given point domain. these cases, corresponding problem optimal recovery approximate best approximation by bounded linear functionals are solved. simply has been completely investigated previously.