ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY

doi:10.1007/s10473-021-0320-0

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (3): 959-967.doi: 10.1007/s10473-021-0320-0

ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY

钟德光¹, 孟凡宁², 袁文俊²

1. Department of Applied Statistics, Guangdong University of Finance, Guangzhou 510521, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

收稿日期:2020-01-06 修回日期:2020-04-04 出版日期:2021-06-25 发布日期:2021-06-07
通讯作者: Wenjun YUAN E-mail:wjyuan1957@126.com
作者简介:Deguang ZHONG,E-mail:huachengzhon@163.com;Fanning MENG,E-mail:mfnfdbx@163.com
基金资助:
This research was supported by NNSF of China (11701111), NNSFs of Guangdong Province (2016A030310257 and 2015A030313346) and the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars.

ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY

Deguang ZHONG¹, Fanning MENG², Wenjun YUAN²

1. Department of Applied Statistics, Guangdong University of Finance, Guangzhou 510521, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received:2020-01-06 Revised:2020-04-04 Online:2021-06-25 Published:2021-06-07
Contact: Wenjun YUAN E-mail:wjyuan1957@126.com
About author:Deguang ZHONG,E-mail:huachengzhon@163.com;Fanning MENG,E-mail:mfnfdbx@163.com
Supported by:
This research was supported by NNSF of China (11701111), NNSFs of Guangdong Province (2016A030310257 and 2015A030313346) and the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars.

摘要/Abstract

摘要： Let $f$ be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality $|\Delta f(z)|\leq B\cdot|D f(z)|^{2}$ for some $B>0$ and $f(0)=0.$ In this note, we show that $f$ does not always satisfy the Schwarz-Pick type inequality $$\frac{1-|z|^{2}}{1-|f(z)|^{2}}\leq C(B),$$ where $C(B)$ is a constant depending only on $B.$ Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions.

关键词: Schwarz-Pick inequality, Poisson differential inequality, hyperbolically Lipschitz continuity

Abstract: Let $f$ be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality $|\Delta f(z)|\leq B\cdot|D f(z)|^{2}$ for some $B>0$ and $f(0)=0.$ In this note, we show that $f$ does not always satisfy the Schwarz-Pick type inequality $$\frac{1-|z|^{2}}{1-|f(z)|^{2}}\leq C(B),$$ where $C(B)$ is a constant depending only on $B.$ Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions.

Key words: Schwarz-Pick inequality, Poisson differential inequality, hyperbolically Lipschitz continuity

中图分类号:

30C80

钟德光, 孟凡宁, 袁文俊. ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY[J]. 数学物理学报(英文版), 2021, 41(3): 959-967.

Deguang ZHONG, Fanning MENG, Wenjun YUAN. ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY[J]. Acta mathematica scientia,Series B, 2021, 41(3): 959-967.

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ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY

ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY

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