Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

Jianfei Wang

doi:10.1155/2012/647972

Abstract

LetBXbe the unit ball in a complex Banach spaceX. AssumeBXis homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ballBntoBXassociated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

Highlights

By the classical Pick’s invariant form of Schwarz’s lemma, a holomorphic function f z which is bounded by one in the unit disk D ⊂ C satisfies the following inequlity1− f z 2 f z ≤ 1 − |z|21.1 at each point z of D
Our result shows that the high-order Schwarz-Pick estimates on the unit ball do depend on the geometric property of the image domain BX
Let Aut Ω denote the set of biholomorphic mappings of Ω onto itself

Summary

Introduction

By the classical Pick’s invariant form of Schwarz’s lemma, a holomorphic function f z which is bounded by one in the unit disk D ⊂ C satisfies the following inequlity1− f z 2 f z ≤ 1 − |z|21.1 at each point z of D. Suppose f z is holomorphic mapping from Bn to Bm. for any multiindex k ≥ 1 and β ∈ Cn \ {0}, Hf z Dk f, z, β , Dk f, z, β ≤ k! We will extend Theorem A to holomorphic mappings from the unit ball When BX Bm, our result coincides with Theorem A.

Results

Conclusion