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

  • Ian Graham
  • Hidetaka Hamada
  • Gabriela Kohr
  • Mirela Kohr
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 26)

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

Primary 32H02; Secondary 30C45 

Notes

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Ian Graham
    • 1
  • Hidetaka Hamada
    • 2
  • Gabriela Kohr
    • 3
  • Mirela Kohr
    • 3
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Faculty of Science and EngineeringKyushu Sangyo UniversityHigashi-ku, FukuokaJapan
  3. 3.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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