Abstract
We consider rigidity of holomorphic map**s from local and global viewpoints. For instance, we derive a generalization of the famous multi-point Schwarz–Pick lemma of Alan Frank Beardon and Carl David Minda that contains a number of known variations of the classical Schwarz lemma. Such global conclusions will be reached by using comparison with proper holomorphic map**s.
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Notes
Although the proofs of this intuitively obvious claim, the reverse implication of which is easy to prove, are undoubtedly less simple, they might be surprisingly hard to find in the literature. For convenience, we mention that a (more general) statement and proof of the differential-geometric version is in Chapter 6 of the book by Lee [12]. (See Problem 6–7 in [12]).
References
Abate, M.: Holomorphic dynamics on hyperbolic Riemann surfaces. In: De Gruyter Studies in Mathematics, vol. 89. De Gruyter, Berlin (2023)
Ahlfors, L.V.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43(3), 359–364 (1938)
Beardon, A.F., Minda, D.: A multi-point Schwarz–Pick lemma. J. Anal. Math. 92, 81–104 (2004)
Baribeau, L., Rivard, P., Wegert, E.: On hyperbolic divided differences and the Nevanlinna–Pick problem. Comput. Methods Funct. Theory 9(2), 391–405 (2009)
Carathéodory, C.: Theory of Functions of a Complex Variable, Vol. 2. Translated by F. Steinhardt. Chelsea Publishing Company, New York (1954)
Cho, K., Kim, S., Sugawa, T.: On a multi-point Schwarz–Pick lemma. Comput. Methods Funct. Theory 12(2), 483–499 (2012)
Ito, M.: Inequalities of Schwarz–Pick type with branch points and the pseudo-hyperbolic distance. Complex Var. Elliptic Equ. 62(6), 786–794 (2017)
Ito, M.: Equality statements for Schwarz–Pick type estimates and the pseudo-hyperbolic distance. Monatsh. Math. 184(1), 105–113 (2017)
Ito, M.: Contracting properties of bounded holomorphic functions. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 32(1), 135–147 (2021)
Julia, G.: Extension nouvelle d’un lemme de Schwarz. Acta Math. 42(1), 349–355 (1920)
Landau, E., Valiron, G.: A deduction from Schwarz’s lemma. J. Lond. Math. Soc. 4(3), 162–163 (1929)
Lee, J.M.: Introduction to Riemannian manifolds. In: Graduate Texts in Mathematics, vol. 176, Second edition of [MR1468735]. Springer, Cham (2018)
Mercer, P.R.: On a strengthened Schwarz–Pick inequality. J. Math. Anal. Appl. 234(2), 735–739 (1999)
Nehari, Z.: A generalization of Schwarz’ lemma. Duke Math. J. 14, 1035–1049 (1947)
Rivard, P.: Some applications of higher-order hyperbolic derivatives. Complex Anal. Oper. Theory 7(4), 1127–1156 (2013)
Rivard, P.: A Schwarz–Pick theorem for higher-order hyperbolic derivatives. Proc. Am. Math. Soc. 139(1), 209–217 (2011)
Wolff, J.: Sur une généralisation d’un théorème de Schwarz. C. R. Acad. Sci. Paris 182, 918–920 (1926)
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The author is grateful to an anonymous referee for precious comments and suggestions that improved the exposition of this paper.
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Ito, M. Rigidity of holomorphic map**s from local and global viewpoints. Annali di Matematica 203, 317–330 (2024). https://doi.org/10.1007/s10231-023-01364-5
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DOI: https://doi.org/10.1007/s10231-023-01364-5