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 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:
$$ |f(z_0)| \le \mathcal{C}\, \|f\|^{\alpha}_{L^{q}_{\varphi_1}(\gamma_1)}\, \|f\|^{\beta}_{L^{p}_{\varphi_0}(\gamma_0)},\quad z_0\in G, \quad 0<q, p\le\infty.$$
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 $1\le q,p\le\infty$, 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.