A Description Method in the Value Region Problem

Prokhorov, Dmitri; Samsonova, K. A.

doi:10.1007/s11785-016-0551-6

Cited by 2 publications

(1 citation statement)

References 5 publications

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“…In [35], the authors described the value region {f (z 0 )} in S R (M ) in an extremely implicit form. They proved that the set {f (z 0 )} in S R (M ) is bounded by the two simple curves γ 1 and γ 2 .…”

Section: Reachable Sets For Univalent Functions With Real Taylor Coefmentioning

confidence: 99%

Value Regions in Classes of Conformal Mappings

Prokhorov¹

2019

MMI

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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 → 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 function families under consideration. The coefficient value regions {(a 2 , . . . , a n ) : f (z) = z + ∞ n=2 a n z n }, |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.Благодарности. Исследование выполнено при финансовой поддержке Российского научного фонда (проект № 17-11-01229). Образец для цитирования:Prokhorov D. V. Value Regions in Classes of Conformal Mappings [Прохоров Д. В. Области значений в классах конформных отображений] // Изв. Сарат. ун-та. Нов. сер.

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Section: Reachable Sets For Univalent Functions With Real Taylor Coefmentioning

confidence: 99%

Value Regions in Classes of Conformal Mappings

Prokhorov¹

2019

MMI

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Three value ranges for symmetric self-mappings of the unit disc

Koch

Schleißinger

2016

Proc. Amer. Math. Soc.

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Let D be the unit disc and z 0 ∈ D. We determine the value range {f (z 0 ) | f ∈ R ≥ }, where R ≥ is the set of holomorphic functions f : D → D with f (0) = 0 and f (0) ≥ 0 that have only real coefficients in their power series expansion around 0, and the smaller set {f (z 0 ) | f ∈ R ≥ , f is typically real}. Furthermore, we describe a third value range {f (z 0 ) | f ∈ I}, where I consists of all univalent self-mappings of the upper half-plane H with hydrodynamical normalization which are symmetric with respect to the imaginary axis.

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A Description Method in the Value Region Problem

Cited by 2 publications

References 5 publications

Value Regions in Classes of Conformal Mappings

Value Regions in Classes of Conformal Mappings

Three value ranges for symmetric self-mappings of the unit disc

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