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

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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© Springer International Publishing AG, part of Springer Nature 2017

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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