Extremum problems for Golubev sums (Q1974364)

For \(s>1\), let \(D_1,D_2,\dots, D_s\) be bounded Jordan domains with disjoint closures and with rectifiable boundaries in the complex plane \(\mathbb{C}\), and let \(\gamma_j= \partial D_j\), \(1\leq j\leq s\). Let \(D= D_1\cup D_2\cup\cdots\cup D_s\), \(G= \mathbb{C}\setminus D\), and \(\gamma= \gamma_1\cup\gamma_2\cup\cdots\cup \gamma_s\). For a set \(A\), let \({\mathcal M}_A\) denote the set of all complex valued measures on \(A\). For \(1\leq j\leq s\), let \(F_j\) be a compact subset of \(D_j\), and let \(\Gamma\) denote the class of all functions \(f\) such that \[ f(z)= \sum^\infty_{j=1} \int_{\gamma_j} {d\lambda_j(t)\over t-z},\quad d\lambda_j\in{\mathcal M}_{F_j}. \] These sums are the Golubev sums of the title. The function \(f\) is analytic in \(G\) and \(|f(z)|\leq 1\) for \(z\in G\). Let \(\{n_j: 1\leq j\leq s\}\) be a sequence of positive integers, let \(d\mu_0\) be a complex measure on \(\gamma\), and let \[ \omega_j(t)= \int_\gamma {d\mu_0(\zeta)\over(t- \zeta)^{n_j}},\quad t\in F_j,\quad 1\leq j\leq s. \] The author proves that the number \[ \beta= \sup\Biggl\{\Biggl|\sum^s_{j=1} \int_{F_j} \omega_j(t) d\lambda_j\Biggr|: \lambda_j\in{\mathcal M}_{F_j}\Biggr\}= \sup_{f\in\Gamma} \Biggl|\int_\gamma f(\zeta) d\mu_0(\zeta)\Biggr|, \] is the solution of a number of other extremal problems involving other classes of functions. Proofs are obtained by duality methods.