Growth and two-point distortion for biholomorphic mappings of the ball

Abstract

In this article we obtain invariant two-point distortion theorems for univalent holomorphic mappings of the unit ball in one and in several variables. A new locally uniform minimum growth bound for biholomorphic mappings of the unit ball in dimension n>1 is obtained for those mappings with finite “norm order”. The results are framed in the context of linear invariant families and the Koebe transform.

¶Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday

Keywords:

• Biholomorphic mapping
• Linear invariant family
• Norm order of a LIF
• Distortion
• Min growth bound
• Hyperbolic metric
• Carathéodory distance

Keywords:

• 1991 Mathematics Subject Classifications: 32H02
• 30C45

Acknowledgement

The authors thank the referee for a very careful reading of the article and for stimulating suggestions.

Notes

¶Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday