Articles

A SCHWARZ LEMMA FOR HARMONIC FUNCTIONS IN THE REAL UNIT BALL

  • Shaoyu DAI ,
  • Huaihui CHEN
Expand
  • 1. School of Mathematics, Southeast University, Nanjing 210096, China Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China;
    2. Department of Mathematics, Nanjing Normal University, Nanjing 210097, China

Received date: 2018-01-02

  Revised date: 2018-12-21

  Online published: 2019-11-11

Supported by

Research supported by the National Natural Science Foundation of China (11201199; 11071083; 11671361) and Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and Presidents.

Abstract

We establish a precise Schwarz lemma for real-valued and bounded harmonic functions in the real unit ball of dimension n. This extends Chen's Schwarz-Pick lemma for real-valued and bounded planar harmonic mapping.

Cite this article

Shaoyu DAI , Huaihui CHEN . A SCHWARZ LEMMA FOR HARMONIC FUNCTIONS IN THE REAL UNIT BALL[J]. Acta mathematica scientia, Series B, 2019 , 39(5) : 1339 -1344 . DOI: 10.1007/s10473-019-0512-z

References

[1] Axler S, Bourdon P, Ramey W. Harmonic Function Theory. Graduate Texts in Mathematics. 2nd ed. New York:Springer-Verlag, 2001
[2] Duren P. Harmonic Mappings in the Plane. Cambridge Tracts in Mathematics, Vol 137. New York:SpringerVerlag, 1992
[3] Bshouty D, Hengartner W. Univalent harmonic mappings in the plane. Ann Univ Mariae Curie-Sklodowska Sect A, 1994, 48:12-42
[4] Chen H, Gauthier P M, Hengartner W. Bloch constants for planar harmonic mappings. Proc Amer Math Soc, 2000, 128:3231-3240
[5] Chen H. The Schwarz-Pick lemma for planar harmonic mappings. Sci China Math, 2011, 54:1101-1118
[6] Heinz E. On one-to-one harmonic mappings. Pacific J Math, 1959, 9:101-105
[7] Chen H. The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings. Sci China Math, 2013, 56:2327-2334
[8] Kalaj D. Harmonic Functions and Harmonic Quasiconformal Mappings Between Convex Domains[D]. Beograd, 2002
[9] Knežević M, Mateljević M. On the quasi-isometries of harmonic quasiconformal mappings. J Math Anal Appl, 2007, 334:404-413
[10] Pavlović M. A Schwarz lemma for the modulus of a vector-valued analiytic function. Proc Amer Math Soc, 2011, 139:969-973
[11] Kalaj D, Vuorinen M. On harmonic functions and the Schwarz lemma. Proc Amer Math Soc, 2012, 140:161-165
[12] Liu Y, Chen Z, Pan Y. A boundary Schwarz lemma for holomorphic mappings on the polydisc. Chin Ann Math, Series B, 2018, 39(1):9-16
[13] Liu Y, Chen Z, Pan Y. Boundary Schwarz lemma for nonequidimensional holomorphic mappings and its application. Pacific J Math, 2018, 295(2):463-476
[14] Liu Y, Dai S, Pan Y. Boundary Schwarz lemma for pluriharmonic mappings between unit balls. J Math Anal Appl, 2016, 433(1):487-495
[15] Chen S, Ponnusamy S, Rasila A, Wang X. Linear connectivity, Schwarz-Pick lemma and univalency criteria for planar harmonic mapping. Acta Math Sin (English Series), 2017, 32(3):297-308
Options
Outlines

/