Acta mathematica scientia,Series A ›› 2017, Vol. 37 ›› Issue (1): 18-25.

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Refinement of Schwarz Lemma for the Modulus of Holomorphic Mappings on the Unit Disk

Zhang Yufang, Xu Qinghua   

  1. School of Science, Zhejiang University of Science and Technology, Hangzhou 310023
  • Received:2016-03-12 Revised:2016-10-22 Online:2017-02-26 Published:2017-02-26
  • Supported by:

    Supported by NSFC (11261022, 11561030), the Jiangxi Provincial Natural Science Foundation (20152ACB20002, 20161BAB201019) and the Natural Science Foundation of Department of Education of Jiangxi Province (GJJ150301)

Abstract:

Let D be the unit disk in C, Bp={z∈Cn:|zi|p<1}, 1 < < +∞. In this note, it is proved that if fHm(D, Bp), then |▽||f||(z)|≤(m|z|m-1)/(1-|z|2m)(1-||f(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: Holomorphic mappings, Zero of order m, Schwarz lemma

CLC Number: 

  • O174.56
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