A description method in the value region problem
type: Item
instance of: scholarly article
title: A description method in the value region problem
zbMATH Open document ID: 1388.30025
author: Kristina Samsonova
author: Dmitri V. Prokhorov
published in: Complex Analysis and Operator Theory
publication date: 2017-12-01T00:00:00Z
review text: 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.
reviewed by: Bodo Dittmar
Mathematics Subject Classification ID: 30C75
Mathematics Subject Classification ID: 30C35
zbMATH DE Number: 6814631
zbMATH Keywords: value region
zbMATH Keywords: Loewner differential equation
MaRDI profile type: Publication
full work available at URL: s11785-016-0551-6
OpenAlex ID: W2304534717
cites work: Classical and Stochastic Löwner–Kufarev Equations
cites work: Q3345031
cites work: On univalent functions with real coefficients
cites work: Q3204714
cites work: Q4305242
cites work: Value range of solutions to the chordal Loewner equation
cites work: Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen
cites work: Rogosinski's lemma for univalent functions, hyperbolic Archimedean spirals and the Loewner equation
cites work: Extremum problems for bounded univalent functions
DOI: 10.1007/S11785-016-0551-6
description: scientific article; zbMATH DE number 6814631