An analogue of the two-constants theorem and optimal recovery of analytic functions

Several related extremal problems for analytic functions in a simply connected domain $G$ with rectifiable Jordan boundary $\Gamma$ are treated. The sharp inequality
$$ |f(z)|\le\mathscr C^{r,q}(z;\gamma_0,\varphi_0;\gamma_1,\varphi_1)\|f\|^\alpha_{L^q_{\varphi_1}(\gamma_1)}\|f\|^{1-\alpha}_{L^r_{\varphi_0}(\gamma_0)} $$
is established between a value of an analytic function in the domain and the weighted integral norms of the restrictions of its boundary values to two measurable subsets $\gamma_1$ and $\gamma_0=\Gamma\setminus\gamma_1$ of the boundary of the domain. It is an analogue of the F. and R. Nevanlinna two-constants theorem. The corresponding problems of optimal recovery of a function from its approximate boundary values on $\gamma_1$ and of the best approximation to the functional of analytic extension of a function from the part of the boundary $\gamma_1$ into the domain are solved.
Bibliography: 35 titles.