Value ranges of univalent self-mappings of the unit disc

Koch, Julia; Schleißinger, Sebastian

doi:10.1016/j.jmaa.2015.08.068

Cited by 6 publications

(6 citation statements)

References 3 publications

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“…The three following theorems belong to Koch and Schleissinger [31]. Due to rotations in S T , a choice of z 0 ∈ D \ {0} in the value region problem for V T (z 0 ) is reduced to z 0 ∈ (0, 1).…”

Section: Reachable Sets For the Radial Loewner Equationmentioning

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 the Radial Loewner Equationmentioning

confidence: 99%

Value Regions in Classes of Conformal Mappings

Prokhorov¹

2019

MMI

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“…One can obtain a refinement of the main result of [RS14] by considering the normalization f (0) = e −T , T > 0, instead of f (0) > 0; see [KS16].…”

Section: Univalent Self-mappings With Real Coefficientsmentioning

confidence: 99%

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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“…6], and references therein. We also mention recent studies [30,38,40], which make essential use of Theorem A and its analogue for hydrodynamically normalized univalent self-maps of the upper half-plane. Finally, it is worth mentioning that the univalence comes to de Branges' proof of Bieberbach's famous conjecture [12] solely via a slight modification of Theorem A.…”

Section: Introductionmentioning

confidence: 99%

Parametric Representations and Boundary Fixed Points of Univalent Self-Maps of the Unit Disk

Gumenyuk

2017

Springer INdAM Series

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A classical result in the theory of Loewner's parametric representation states that the semigroup U * of all conformal self-maps φ of the unit disk D normalized by φ(0) = 0 and φ ′ (0) > 0 can be obtained as the reachable set of the Loewner -Kufarev control systemwhere the control functions t → G t ∈ Hol(D, C) form a certain convex cone. Here we extend this result to the semigroup U[F ] consisting of all conformal φ : D → D whose set of boundary regular fixed points contains a given finite set F ⊂ ∂D and to its subsemigroup U τ [F ] formed by id D and all φ ∈ U[F ]\ {id D } with the prescribed boundary Denjoy -Wolff point τ ∈ ∂D \ F . This completes the study launched in [29], where the case of interior Denjoy -Wolff point τ ∈ D was considered. 2010 Mathematics Subject Classification. Primary: 30C35, 30C75; Secondary: 30D05, 30C80, 34H05, 37C25.

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Value ranges of univalent self-mappings of the unit disc

Abstract: We describe the value set {f (z 0 ) :, where D denotes the unit disc and z 0 ∈ D \ {0}, T > 0, by applying Pontryagin's maximum principle to the radial Loewner equation.

Cited by 6 publications

References 3 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

Parametric Representations and Boundary Fixed Points of Univalent Self-Maps of the Unit Disk

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