A refinement of the Schwarz-Pick estimates and the Carathéodory metric in several complex variables

Abstract

In this article, we first establish an asymptotically sharp result on the higher order Fréchet derivatives for bounded holomorphic mappings $f(x) = f(0) + \sum\limits_{s = 1}^\infty {{{{D^{sk}}f(0)({x^{sk}})} \over {(sk)!}}} :\,{B_X} \to {B_Y}$, where B_X is the unit ball of X. We next give a sharp result on the first order Fréchet derivative for bounded holomorphic mappings $f(x) = f(0) + \sum\limits_{s = k}^\infty {{{{D^s}f(0)({x^s})} \over {s!}}} :\,{B_X} \to {B_Y}$, where B_X is the unit ball of X. The results that we derive include some results in several complex variables, and extend the classical result in one complex variable to several complex variables.