Extremal Problems and Convergence Results for Mappings with Generalized Parametric Representation in ℂn

  • Hidetaka Hamada
  • Mihai Iancu
  • Gabriela KohrEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 26)


In this paper we survey recent results related to extremal problems for the family \(\widetilde {S}^t_A(\mathbb {B}^n)\) of normalized univalent mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\), which have generalized parametric representation with respect to time-dependent operators \(A\in \skew 4\widetilde {\mathcal {A}}\), where \(\skew 4\widetilde {\mathcal A}\) is the family of all measurable mappings \(A: [0,\infty )\to L(\mathbb {C}^n)\), which satisfy certain natural conditions. In the second part of this paper, we consider the dependence of \(\widetilde {S}^t_A(\mathbb {B}^n)\) on t ≥ 0 and on \(A\in \skew 4\widetilde {\mathcal {A}}\), and we present some convergence results related to the family \(\widetilde {S}^t_A(\mathbb {B}^n)\) in terms of the Carathéodory metric ρ on \(H(\mathbb {B}^n)\). Various questions and remarks are also provided, which point out main differences between the usual parametric representation with respect to time-independent operators and that with respect to time-dependent operators.


Carathéodory family Extreme point Generalized parametric representation Loewner chain Loewner differential equation Support point 

2000 Mathematics Subject Classification

Primary 32H02; Secondary 30C45 



H. Hamada is partially supported by JSPS KAKENHI Grant Number JP16K05217.


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Authors and Affiliations

  1. 1.Faculty of Science and EngineeringKyushu Sangyo UniversityHigashi-ku FukuokaJapan
  2. 2.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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