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Computational Methods and Function Theory

, Volume 7, Issue 2, pp 379–399 | Cite as

Koebe Invariant Functions and Extremal Problems for Holomorphic Mappings in the Unit Ball of ℂn

  • John A. Pfaltzgraff
  • Ted J. Suffridge
Article

Abstract

This work is concerned with locally biholomorphic mappings on the unit ball of complex n-dimensional space. We define a concept called K-invariance and give a complete characterization of K-invariant functions. We also explore the connection between K-invariance and the solution of extremal problems in certain linear invariant families.

Keywords

Locally biholomorphic mapping linear invariance invariant function extremal function 

2000 MSC

32H02 30C45 30C55 

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References

  1. 1.
    C. H. FitzGerald, Invariant mappings in geometric function theory, Proceedings Satellite Conference to International Congress of Mathematicians in Beijing 2002, World Scientific, 2004, 118–122.Google Scholar
  2. 2.
    I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York, 2003.MATHGoogle Scholar
  3. 3.
    I. Graham and G. Kohr, Univalent mappings associated with the Roper-Suffridge extension operator, J. Anal. Math. 81 (2000), 331–342.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    J. R. Muir, Jr., A modification of the Roper-Suffridge extension operator, Comput. Methods Funct. Theory 5 (2005) no.1, 237–251.MathSciNetMATHGoogle Scholar
  5. 5.
    J. R. Muir, Jr. and T. J. Suffridge, Some extreme points of the family of normalized convex mappings of the unit ball of ℂn, J. Anal. Math. 98 (2006), 169–182.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    J. A. Pfaltzgraff and T. J. Suffridge, Invariant mappings under Koebe transforms of the ball, # 939-32-821, AMS Abstracts vol. 20, 1 (1999) 75.Google Scholar
  7. 7.
    J. A. Pfaltzgraff and T. J. Suffridge, An extension theorem and linear invariant families generated by starlike maps, Ann. Univ. Mariae Curie Sklodowska, Sect. A 53 (1999), 193–207.MathSciNetMATHGoogle Scholar
  8. 8.
    J. A. Pfaltzgraff and T. J. Suffridge, Norm order and geometric properties of holomorphic mappings in ℂn, J. Anal. Math. 82(2000), 285–313.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    C. Pommerenke, Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155 (1964), 108–154.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    C. Pommerenke, Linear-invariante Familien analytischer Funktionen II, Math. Ann. 156 (1964), 226–262.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    K. Roper and T. J. Suffridge, Convex mappings of the unit ball of ℂn, J. Anal. Math. 65(1995), 333–347.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Heldermann  Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of North CarolinaChapel HillUSA
  2. 2.Department of MathematicsUniversity of KentuckyLexingtonUSA

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