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Erschienen in:
The Embedding Conjecture and the Approximation Conjecture in Higher Dimension Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

Loewner Chains and Extremal Problems for Mappings with A-Parametric Representation in ℂ n

verfasst von : Ian Graham, Hidetaka Hamada, Gabriela Kohr, Mirela Kohr

Erschienen in: Geometric Function Theory in Higher Dimension

Verlag: Springer International Publishing

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Abstract

In this paper we survey various results concerning extremal problems related to Loewner chains, the Loewner differential equation, and Herglotz vector fields on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). First, we survey recent results related to extremal problems for the Carathéodory families \({\mathcal M}\) and \({\mathcal N}_A\) on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), where \(A\in L(\mathbb {C}^n)\) with m(A) > 0. In the second part of this paper, we present recent results related to extremal problems for the family \(S_A^0(\mathbb {B}^n)\) of normalized univalent mappings with A-parametric representation on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), where \(A\in L(\mathbb {C}^n)\) with k +(A) < 2m(A). In the last section we survey certain results related to extreme points and support points for a special compact subset of \(S_A^0(\mathbb {B}^n)\) consisting of bounded mappings on \(\mathbb {B}^n\). Particular cases, open problems, and questions will be also mentioned.

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Vorheriges Kapitel On a Solution of a Particular Case of Aliaga-Tuneski Question
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Metadaten
Titel
Loewner Chains and Extremal Problems for Mappings with A-Parametric Representation in ℂ n
verfasst von
Ian Graham
Hidetaka Hamada
Gabriela Kohr
Mirela Kohr
Copyright-Jahr
2017
Buch
Geometric Function Theory in Higher Dimension

Print ISBN: 978-3-319-73125-4

Electronic ISBN: 978-3-319-73126-1

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-73126-1_13