Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
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Prokhorov D. V. Value Regions in Classes of Conformal Mappings. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 3, pp. 258-279. DOI: 10.18500/1816-9791-2019-19-3-258-279

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2019
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517.54

Value Regions in Classes of Conformal Mappings

Autors: 
Prokhorov Dmitri Valentinovich, Saratov State University
Abstract: 

The survey is devoted to most recent results in the value region problem over different classes of holomorphic univalent functions represented by solutions to the Loewner differential equations both in the radial and chordal versions. It is important also to present classical and modern solution methods and to compare their efficiency. More details are concerned with optimization methods and the Pontryagin maximum principle, in particular. A value region is the set {f(z 0 )} of all possible values for the functional f 7→ f(z 0 ) where z 0 is a fixed point either in the upper half-plane for the chordal case or in the unit disk for the radial case, and f runs through a class of conformal mappings. Solutions to the Loewner differential equations form dense subclasses of functio families under consideration. The coefficient value regions {(a 2 ,...,a n ) : f(z) = z+P ∞n=2 a n zn }, |z| < 1, are the part of the field closely linked with extremal problems and the Bombieri conjecture about the structure of the coefficient region for the class S in a neighborhood of the point (2,...,n) corresponding to the Koebe function.

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Received: 
07.04.2018
Accepted: 
12.05.2019
Published: 
31.08.2019