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

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DOI: 10.1007/BF02791231

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

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## Abstract

We introduce a new notion of the order of a linear invariant family of locally biholomorphic mappings on the*n*-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.