Journal d’Analyse Mathématique

, Volume 82, Issue 1, pp 285–313

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


    • Department of MathematicsUniversity of North Carolina
  • T. J. Suffridge
    • Department of MathematicsUniversity of Kentucky

DOI: 10.1007/BF02791231

Cite this article as:
Pfaltzgraff, J.A. & Suffridge, T.J. J. Anal. Math. (2000) 82: 285. doi:10.1007/BF02791231


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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© Hebrew University of Jerusalem 2000