Abstract
The authors prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary Schwarz lemma.
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References
Ruscheweyh, S., Two remarks on bounded analytic functions, Serdica, 11(2), 1985, 200–202.
Dai, S. and Pan, Y., Note on Schwarz-Pick estimates for bounded and positive real part analytic functions, Proceedings of the American Mathematical Society, 136(2), 2008, 635–640.
Rudin, W., Function Theory in Polydiscs, W. A. Benjamin, New York, 1969.
Knese, G., A Schwarz lemma on the polydisk, Proceedings of the American Mathematical Society, 135(9), 2007, 2759–2768.
Liu, Y. and Chen, Z., Schwarz-Pick estimates for holomorphic mappings from the polydisk to the unit ball, Journal of Mathematical Analysis and Applications, 376(1), 2011, 123–128.
Garnett, J., Bounded Analytic Functions, Academic Press, New York, 1981.
Krantz, S., The Schwarz lemma at the boundary, Complex Variables and Elliptic Equations, 56(5), 2011, 455–468.
Liu, T., Wang, J. and Tang, X., Schwarz lemma at the boundary of the unit ball in Cn and its applications, Journal of Geometry Analysis, 25(3), 2015, 1890–1914.
Tang, X., Liu, T. and Lu, J., Schwarz lemma at the boundary of the unit polydisk in Cn, Science China Mathematics, 58(8), 2015, 1639–1652.
Liu, Y., Dai, S. and Pan, Y., Boundary Schwarz lemma for pluriharmonic mappings between unit balls, Journal of Mathematical Analysis and Applications, 433(1), 2016, 487–495.
Chen, Z., Liu, Y. and Pan Y., A Schwarz lemma at the boundary of Hilbert balls, Chin. Ann. Math. Ser. B, to appear.
Alexander, H., Holomorphic mappings from the ball and polydisc, Mathematische Annalen, 209(3), 1974, 249–256.
Alexander, H., Extremal holomorphic imbeddings between the ball and polydisc, Proceedings of the American Mathematical Society, 68(2), 1978, 200–202.
Jarnicki, M. and Pflug, P., Invariant distances and metrics in complex analysis, Walter de Gruyter, 9, Berlin, 1993.
Rudin, W., Function Theory in the Unit Ball of Cn, Springer-Verlag, New York, 2009.
Kobayashi, S., Intrinsic metrics on complex manifolds, Bulletin of the American Mathematical Society, 73(3), 1967, 347–349.
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This work was supported by the National Science Foundation of China (Nos. 11671361, 11571256).
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Liu, Y., Chen, Z. & Pan, Y. A boundary Schwarz lemma for holomorphic mappings on the polydisc. Chin. Ann. Math. Ser. B 39, 9–16 (2018). https://doi.org/10.1007/s11401-018-1047-7
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DOI: https://doi.org/10.1007/s11401-018-1047-7