A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES

doi:10.1007/s10473-024-0409-3

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (4): 1337-1346.doi: 10.1007/s10473-024-0409-3

A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES

Xiaosong liu^1,*, Taishun liu²

1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China;
2. Department of Mathematics, Huzhou University, Huzhou 313000, China

收稿日期:2022-11-20 修回日期:2023-01-08 出版日期:2024-08-25 发布日期:2024-08-30

A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES

Xiaosong liu^1,*, Taishun liu²

1. School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China;
2. Department of Mathematics, Huzhou University, Huzhou 313000, China

Received:2022-11-20 Revised:2023-01-08 Online:2024-08-25 Published:2024-08-30
Contact: *E-mail: lxszhjnc@163.com
About author:E-mail: tsliu@hutc.zj.cn
Supported by:
The first author's research was supported by the NSFC (11871257, 12071130) and the second author's research was supported by the NSFC (11971165).

摘要/Abstract

摘要： In this article, we first establish an asymptotically sharp result on the higher order Fréchet derivatives for bounded holomorphic mappings $f(x)=f(0)+\sum\limits_{s=1}^\infty\frac{D^{sk} f(0)(x^{sk})}{(sk) !}: B_X\rightarrow B_Y$, 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 $f(x)=f(0)+\sum\limits_{s=k}^\infty\frac{D^{s} f(0)(x^{s})}{s !}: B_X\rightarrow B_Y$, 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.

关键词: refined Schwarz-Pick estimate, bounded holomorphic mapping, Carathéodory metric, first order Fréchet derivative, higher order Fréchet derivatives

Abstract: In this article, we first establish an asymptotically sharp result on the higher order Fréchet derivatives for bounded holomorphic mappings $f(x)=f(0)+\sum\limits_{s=1}^\infty\frac{D^{sk} f(0)(x^{sk})}{(sk) !}: B_X\rightarrow B_Y$, 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 $f(x)=f(0)+\sum\limits_{s=k}^\infty\frac{D^{s} f(0)(x^{s})}{s !}: B_X\rightarrow B_Y$, 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.

Key words: refined Schwarz-Pick estimate, bounded holomorphic mapping, Carathéodory metric, first order Fréchet derivative, higher order Fréchet derivatives

中图分类号:

32A30

Xiaosong liu, Taishun liu. A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES[J]. 数学物理学报(英文版), 2024, 44(4): 1337-1346.

Xiaosong liu, Taishun liu. A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES[J]. Acta mathematica scientia,Series B, 2024, 44(4): 1337-1346.

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A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES

A REFINEMENT OF THE SCHWARZ-PICK ESTIMATES AND THE CARATHÉODORY METRIC IN SEVERAL COMPLEX VARIABLES

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