Boundary Schwarz Lemma of the Unit Ball in Rn Satisfying Poisson's Equation

Ming Xin CHEN, Jian Fei WANG, Xiao Min TANG

Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (4) : 717-726.

PDF(479 KB)
PDF(479 KB)
Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (4) : 717-726. DOI: 10.12386/b20210679

Boundary Schwarz Lemma of the Unit Ball in Rn Satisfying Poisson's Equation

  • Ming Xin CHEN1, Jian Fei WANG1, Xiao Min TANG2
Author information +
History +

Abstract

Schwarz lemma plays significant roles on function theory of holomorphic mappings or harmonic mappings. In this paper, we establish the Schwarz lemma at the boundary for self-mappings solutions satisfying the Poisson's equation of the unit ball in Rn. As an application, the boundary Schwarz lemma for harmonic self-mappings on the unit ball is obtained which extends the result of pluriharmonic mappings to harmonic mappings in higher dimensions.

Key words

harmonic mapping / boundary Schwarz lemma / Poisson’s equation

Cite this article

Download Citations
Ming Xin CHEN, Jian Fei WANG, Xiao Min TANG. Boundary Schwarz Lemma of the Unit Ball in Rn Satisfying Poisson's Equation. Acta Mathematica Sinica, Chinese Series, 2023, 66(4): 717-726 https://doi.org/10.12386/b20210679

References

[1] Axler S., Bourdon P., Ramey W., Harmonic Function Theory, Graduate Texts in Mathematics, Vol.137, Springer, New York, 1992.
[2] Bonk M., On Bloch’s constant, Proc. Amer. Math. Soc., 1990, 110(4): 889-894.
[3] Burns D., Krantz S., Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc., 1994, 7: 661-676.
[4] Chelst D., A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc., 2001, 129(11): 3275- 3278.
[5] Chen H. H., Gauthier P., The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings, Proc. Amer. Math. Soc., 2011, 139: 583-595.
[6] Chen S. L., Kalaj D., The Schwarz type lemmas and the Landau type theorem of mappings satisfying Poisson’s equations, Complex Anal. Oper. Theory, 2019, 13(4): 2049-2068.
[7] Chen S. L., Ponnusamy S., Schwarz lemmas for mappings satisfying Poisson’s equation, Indag. Math. (N.S.), 2019, 30(6): 1087-1098.
[8] Chen S. L., Vuorinen M., Some properties of a class of elliptic partial differential operators, J. Math. Anal. Appl., 2015, 431: 1124-1137.
[9] Erdélyi A., Magnus W., et al., Higher Transcendental Functions, McGraw-Hill, New York, 1953.
[10] Garnett J., Bounded Analytic Functions, Academic Press, New York, 1981.
[11] Graham I., Hamada H., Kohr G., A Schwarz lemma at the boundary on complex Hilbert balls and applications to starlike mappings, J. Anal. Math., 2020, 140: 31-53.
[12] Harnack A., Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, Leipzig: B. G. Teubner, 1887.
[13] Huang X. J., A boundary rigidity problem for holomorphic mappings on some weakly pseudoconvex domains, Canad. J. Math., 1995, 47(2): 405-420.
[14] Huang X. J., On a semi-rigidity property for holomorphic maps, Asian J. Math., 2003, 7(4): 463-492.
[15] Kalaj D., Heinz-Schwarz inequalities for harmonic mappings in the unit ball, Ann. Acad. Sci. Fenn. Math., 2016, 41: 457-464.
[16] Kalaj D., Schwarz lemma for harmonic mappings in the unit ball, Complex Anal. Oper. Theory, 2018, 12(2): 545-554.
[17] Kalaj D., Pavlovic M., On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation, Trans. Amer. Math. Soc., 2011, 363(8): 4043-4061.
[18] Kalaj D., Vuorinen M, On harmonic functions and the Schwarz lemma, Proc. Amer. Math. Soc., 2012, 140(1): 161-165.
[19] Kim K., Lee H., Schwarz’s lemma from a differential geometric viewpoint, Bangalore, IISc Press, 2011.
[20] Krantz S., The Schwarz lemma at the boundary, Complex Var. Elliptic Equ., 2011, 56(5): 455-468.
[21] Liu T. S., Tang X. M., Schwarz lemma at the boundary of strongly pseudoconvex domain in Cn, Math. Ann., 2016, 366: 655-666.
[22] Liu T. S., Tang X. M., A boundary Schwarz lemma on the classical domain of type (I), Sci. China Math., 2017, 60(7): 1239-1258.
[23] Liu T. S., Tang X. M., Zhang W. J., Schwarz lemma at the boundary on the classical domain of type (III), Chin. Ann. Math. Ser. B, 2020, 41(3): 335-360.
[24] Liu T. S., Ren G. B., The growth theorem of convex mappings on bounded convex circular domains, Science in China, Series A, 1998, 41(2): 123-130.
[25] Liu T. S., Wang J. F., Tang X. M., Schwarz lemma at the boundary of the unit ball in Cn and its applications, J. Geom. Anal., 2015, 25: 1890-1914.
[26] Liu Y. S., Dai S. Y., Pan Y. F., Boundary Schwarz lemma for pluriharmonic mappings between unit balls, J. Math. Anal. Appl., 2016, 433: 487-495.
[27] Osserman R., A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc., 2000, 128: 3513-3517.
[28] Tang X. M., Liu T. S., Zhang W. M., Schwarz lemma at the boundary on the classical domain of type (II), J. Geom. Anal., 2018, 28(2): 1610-1634.
[29] Wang J. F., Liu T. S., Tang X. M., Schwarz lemma at the boundary on the classical domain of type (IV), Pacific J. Math., 2019, 302: 309-333.
[30] Wang X. T., Zhu J. F., Boundary Schwarz lemma for solutions to Poisson’s equation, J. Math. Anal. Appl., 2018, 463: 623-633.
PDF(479 KB)

381

Accesses

0

Citation

Detail

Sections
Recommended

/