The size of the set where a univalent function is large
Let \(S\) denote the set of univalent analytic functions \(f\) in the unit disc \(D\) with \(f(0)=0\), \(f'(0)=1\). It is known that \(f\) has nontangential boundary values \(f(e^{i\theta})\) for a real \(\theta\) a.e. For \(t\in (0,\infty)\), let \[ F(t,f): =\biggl\{e^{i\theta} \in\partial D: \bigl|f(e^{i\theta}) \bigr|<t\biggr\}. \] \textit{A. Solynin} proposed the problem of minimizing, for a fixed \(t\), the one-dimensional Lebesgue measure \(|E(t,f) |\) as \(f\) varies over \(S\). [Zap. Nauchn. Semin. Pomi 204, 115-142 (1993; Zbl 0802.30019)]. The author of the paper being reviewed presents the solving of Solynin's problem.
scientific article; zbMATH DE number 1019377
scientific article