Punishing factors for angles
The authors study the problem of maximizing \(\frac{f^{n}(z_0)}{n!}\) over all not necessarily univalent analytic functions \(f\) mapping some simply connected domain on the Riemann sphere into another, in terms of the Poincaré densities of \(z_0\) in the source domain and \(f(z_0)\) in the target domain, and sometimes also the Poincaré distance from \(z_0\) to \(\infty.\) Here's one of the main results: Suppose that \(f\) maps a simply connnected domain \(\Omega\) in the sphere into some sector \(H\) of given angular opening \(\alpha \pi,\) where \(1\leq \alpha \leq 2.\) Then extremal functions for the \(n\)th derivative problem are furnished by univalent maps of the sphere minus a plane line segment onto \(H.\) Tools in the paper include Taylor series of particular functions, subordination, the validity of Rogosinski's generalization of Bieberbach's conjecture \((| a_n| \leq n\,\) for every function in the unit disk which is subordinate to a function in \(S),\,\) and a generalization to meromorphic functions of Löwner's theorem about coefficients of inverse functions, due to G. Schober and the reviewer.
scientific article; zbMATH DE number 2112931