Journal d’Analyse Mathématique

, Volume 82, Issue 1, pp 285–313

Norm order and geometric properties of holomorphic mappings in\(\mathbb{C}^n \)

Article

Abstract

We introduce a new notion of the order of a linear invariant family of locally biholomorphic mappings on then-ball. This order, which we call the norm order, is defined in terms of the norm rather than the trace of the “second Taylor coefficient operator” of mappings in a family. Sharp bounds on ‖Df(z)‖ and ‖f(z)‖, a general covering theorem for arbitrary LIFs and results about convexity, starlikeness, injectivity and other geometric properties of mappings given in terms of the norm order illustrate the useful nature of this notion. The norm order has a much broader range of influence on the geometric properties of mappings than does the “trace” order that the present authors and many others have used in recent years.

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References

  1. [1]
    R. W. Barnard, C. H. FitzGerald and S. A. Gong,A distortion theorem for biholomorphic mappings in C 2, Trans. Amer. Math. Soc.344 (1994), 907–924.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    D. M. Campbell, J. A. Cima and J. A. Pfaltzgraff,Linear spaces and linear invariant-families of locally univalent functions, Manuscripta Math.4 (1971), 1–30.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    C. H. FitzGerald and C. R. Thomas,Some bounds on convex mappings in several complex variables, Pacific J. Math.165 (1994), 295–320.MATHMathSciNetGoogle Scholar
  4. [4]
    S. Gong,Convex and Starlike Mappings in Several Complex Variables, Vol. 435, Science Press/Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.MATHGoogle Scholar
  5. [5]
    S. Gong,The Bieberbach Conjecture, Amer. Math. Soc., Providence, 1999.MATHGoogle Scholar
  6. [6]
    S. Gong and T. Liu,The growth theorem of biholomorphic convex mappings on B p, Chinese Quart. J. Math.6 (1991), 78–82.Google Scholar
  7. [7]
    I. Graham and G. Kohr,An extension theorem and subclasses of univalent mappings in several complex variables, Complex Variables (to appear).Google Scholar
  8. [8]
    L. Hörmander,On a theorem of Grace, Math Scand.2 (1954), 55–64.MATHMathSciNetGoogle Scholar
  9. [9]
    K. Kikuchi,Starlike and convex mappings in several complex variables, Pacific J. Math.44 (1973), 569–580.MATHMathSciNetGoogle Scholar
  10. [10]
    W. Ma and C. D. Minda,Linear invariance and uniform local univalence, Complex Variables16 (1991), 9–19.MATHMathSciNetGoogle Scholar
  11. [11]
    J. A. Pfaltzgraff,Subordination chains and quasiconformal extension of holomorphic maps in C n, Ann. Acad. Sci. Fenn. Ser. A I Math.1 (1975), 13–25.MathSciNetGoogle Scholar
  12. [12]
    J. A. Pfaltzgraff,Distortion of locally biholomorphic maps of the n-ball, Complex Variables33 (1997), 239–253.MATHMathSciNetGoogle Scholar
  13. [13]
    J. A. Pfaltzgraff and T. J. Suffridge,Linear invariance, order and convex maps in C n, Complex Variables40 (1999), 35–50.MATHMathSciNetGoogle Scholar
  14. [14]
    J. A. Pfaltzgraff and T. J. Suffridge,An extension theorem and linear invariant families generated by starlike maps, Ann. Univ. Mariae Curie-Skłodowska. Sect. A53 (1999), 193–207.MATHMathSciNetGoogle Scholar
  15. [15]
    Chr. Pommerenke,Linear-invarianten Familien analytischer Funktionen I, Math. Ann.155 (1964), 108–154.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    K. A. Roper and T. J. Suffridge,Convexity properties of holomorphic mappings in C n, Trans. Amer. Math. Soc.351 (1999), 1803–1833.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    W. Rudin, Function Theory in the Unit Ball of\(\mathbb{C}^n \), Springer-Verlag, New York, 1980.MATHGoogle Scholar
  18. [18]
    T. J. Suffridge,Starlike and convex maps in Banach spaces, Pacific J. Math.46 (1973), 575–589.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2000

Authors and Affiliations

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

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