The Refined Schwarz-Pick Estimates for Positive Real Part Holomorphic Functions in Several Complex Variables

Abstract

In this article, the refined Schwarz-Pick estimates for positive real part holomorphic functions

$$p(x)=p(0)+\sum\limits_{m=k}^{\infty}{{D^{m}p(0)(x^{m})}\over{m!}}:G\rightarrow\mathbb{C}$$

are given, where k is a positive integer, and G is a balanced domain in complex Banach spaces. In particular, the results of first order Fréchet derivative for the above functions and higher order Fréchet derivatives for positive real part holomorphic functions

$$p(x)=p(0)+\sum\limits_{s=1}^{\infty}{{D^{sk}p(0)(x^{sk})}\over{(sk)!}}:G\rightarrow\mathbb{C}$$

are sharp for G = B, where B is the unit ball of complex Banach spaces or the unit ball of complex Hilbert spaces. Their results reduce to the classical result in one complex variable, and generalize some known results in several complex variables.