A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D

doi:10.1016/S0252-9602(14)60141-7

数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (1): 89-94.doi: 10.1016/S0252-9602(14)60141-7

A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D

刘洋|戴邵虞

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China

收稿日期:2014-01-24 出版日期:2015-01-20 发布日期:2015-01-20
基金资助:
This work was supported by National Natural Science Foundations of China (11011373, 11201199, 11271333) and Zhejiang Provincial Natural Science Foundation of China (LY14A010008).

A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D

LIU Yang, DAI ShaoYu

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China

Received:2014-01-24 Online:2015-01-20 Published:2015-01-20
Supported by:
This work was supported by National Natural Science Foundations of China (11011373, 11201199, 11271333) and Zhejiang Provincial Natural Science Foundation of China (LY14A010008).

摘要/Abstract

摘要：

In this note, we consider a holomorphic mapping f from the unit disk D in C to p-ball B^p ={z ∈ Cⁿ :∑ⁿ _i=1 |z_i|p < 1 , 1 < p < +∞. It is proved that for such f,

|∇||f||(z)| ≤ 1 − ||f(z)||² /1 − |z|² , z ∈ D.

The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.

关键词: Schwarz lemma, holomorphic mapping, p-ball

Abstract:

In this note, we consider a holomorphic mapping f from the unit disk D in C to p-ball Bp = z ∈ Cn :
n Pi=1 |zi|p < 1 , 1 < p < +∞. It is proved that for such f, |∇||f||(z)| ≤ 1 − ||f(z)||2 1 − |z|2 , z ∈ D. The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.

Key words: Schwarz lemma, holomorphic mapping, p-ball

中图分类号:

32H02

刘洋|戴邵虞. A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D[J]. 数学物理学报(英文版), 2015, 35(1): 89-94.

LIU Yang, DAI ShaoYu. A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D[J]. Acta mathematica scientia,Series B, 2015, 35(1): 89-94.

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A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D

A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D

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