Schwarz Lemma at the Boundary for Holomorphic Mappings Between Non-equidimensional Unit Polydiscs

Abstract

In this paper, the authors prove a Schwarz lemma at the boundary for holomorphic mappings between non-equidimensional unit polydiscs and consider an inequality for holomorphic self-mappings of the unit polydisc.

Keywords

Schwarz lemma at the boundary / Unit polydisc / Holomorphic mappings / 30C80 / 30C20

References

[1] Garnett J B. Bounded Analytic Functions, 1981, New York, Academic Press

[2] 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(1): 31-53

[3] Hamada H. A simple proof for the boundary Schwarz lemma for pluriharmonic mappings. Ann. Acad. Sci. Fenn. Math., 2017, 42: 799-802

[4] Hamada H. A Schwarz lemma at the boundary using the Julia-Wolff-Carathéodory type condition on finite dimensional irreducible bounded symmetric domains. J. Math. Anal. Appl., 2018, 465(1): 196-210

[5] Hamada H, Kohr G. A boundary Schwarz lemma for mappings from the unit polydisc to irreducible bounded symmetric domains. Math. Nachrichten, 2020, 293(7): 1345-1351

[6] He L, Tu Z H. The Schwarz lemma at the boundary of the non-convex complex ellipsoids. Acta Math. Sci., 2019, 39(4): 915-926

[7] Krantz S G. Complex Analysis: The Geometric Viewpoint, 2004, Washington, American Mathematical Soc.

[8] Li H-P, Mateljević M. Boundary Schwarz lemma for harmonic and pluriharmonic mappings in the unit ball. J. Math. Inequal., 2022, 16(2): 477-498

[9] Liu T S, Tang X M. A new boundary rigidity theorem for holomorphic self-mappings of the unit ball in ℂⁿ. Pure Appl. Math. Q., 2015, 11(1): 115-130

[10] Liu T S, Tang X M. Schwarz lemma at the boundary of strongly pseudoconvex domains in ℂⁿ. Math. Ann., 2016, 366(1): 655-666

[11] 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

[12] Liu T S, Tang X M. Schwarz lemma and rigidity theorem at the boundary for holomorphic mappings on the unit polydisk in ℂⁿ. J. Math. Anal. Appl., 2020, 489(2): 124148

[13] 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(3): 1890-1914

[14] Liu Y, Chen Z H, Pan Y F. A boundary Schwarz lemma for holomorphic mappings on the polydisc. Chin. Ann. Math. Ser. B, 2018, 39(1): 9-16

[15] Liu Y, Dai S Y, Pan Y F. Boundary Schwarz lemma for pluriharmonic mappings between unit balls. J. Math. Anal. Appl., 2016, 433(1): 487-495

[16] Mateljević M, Mutavdžić N. The boundary schwarz lemma for harmonic and pluriharmonic mappings and some generalizations. Bull. Malays. Math. Sci. Soc., 2022, 45(6): 3177-3195

[17] Osserman R. A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc, 2000, 128(12): 3513-3517

[18] Tang X M, Liu T S, Lu J. Schwarz lemma at the boundary of the unit polydisk in ℂⁿ. Sci. China Math., 2015, 58(8): 1639-1652

[19] Tu Z H, Zhang S. The Schwarz lemma at the boundary of the symmetrized bidisc. J. Math. Anal. Appl., 2018, 459(1): 182-202

[20] Zhu J F. Schwarz lemma and boundary Schwarz lemma for pluriharmonic mappings. Filomat, 2018, 32(15): 5385-5402

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