A description method in the value region problem
The aim of the paper under review is the value region \(\{f(z_0)\}, |z_0| < 1, \) for all bounded holomorphic univalent functions \(f(z) = z + a_2z^2 +\dots \) in the unit disk \(\mathbb{D}\) with \(|f(z)| <M \) and with real Taylor coefficients. For the class of all holomorphic univalent function in the unit disk with real Taylor coefficients the problem was solved by \textit{S. I. Fedorov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 139, 156--167 (1984; Zbl 0552.30009)]. The authors follow \textit{O. Roth} and \textit{S. Schleißinger} [Bull. Lond. Math. Soc. 46, No. 5, 1099--1109 (2014; Zbl 1301.30041)] and reduce the problem using the Loewner differential equation to the optimal control problem for the Hamilton system of controllable differential equations. It follows from the Pontryagin maximum principle that the value region is bounded by a slit domain which is described in details in the paper.
scientific article; zbMATH DE number 6814631