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.
Keywords
- Carathéodory family
- Extreme point
- Herglotz vector field
- Loewner chain
- Loewner differential equation
- Support point
2000 Mathematics Subject Classification
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Acknowledgements
I. Graham was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. H. Hamada was partially supported by JSPS KAKENHI Grant Number JP16K05217.
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Graham, I., Hamada, H., Kohr, G., Kohr, M. (2017). Loewner Chains and Extremal Problems for Mappings with A-Parametric Representation in ℂn . In: Bracci, F. (eds) Geometric Function Theory in Higher Dimension. Springer INdAM Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-73126-1_13
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