Abstract
The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls inC n and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of the main tools for the proof is a general version of the Lindelöf principle valid for not necessarily bounded holomorphic functions.
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References
[A1] M. Abate,Horospheres and iterates of holomorphic maps, Math. Z.198 (1988), 225–238.
[A2] M. Abate,Common fixed points of commuting holomorphic maps, Math. Ann.283 (1989), 645–655.
[A3] M. Abate,The Lindelöf principle and the angular derivative in strongly convex domains, J. Analyse Math.54 (1990), 189–228.
[A4] M. Abate,Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende, 1989.
[A5] M. Abate,Angular derivatives in strongly pseudoconvex domains, Proc. Symp. Pure Math.52, Part 2 (1991), 23–40.
[B] R. B. Burckel,An Introduction to Classical Complex Analysis, Academic Press, New York, 1979.
[C] C. Carathéodory,Über die Winkelderivierten von beschränkten analytischen Funktionen, Sitzungsber. Preuss. Akad. Wiss., Berlin (1929), 39–54.
[C-K] J. A. Cima and S. G. Krantz,The Lindelöf principle and normal functions of several complex variables, Duke Math. J.50 (1983), 303–328.
[Č] E. M. Čirka, The Lindelöf and Fatou theorems inC n, Math. USSR-Sb.21 (1973), 619–641.
[D] P. V. Dovbush,Existence of admissible limits of functions of several complex variables, Sib. Math. J.28 (1987), 83–92.
[D-Z] Ju. N. Drožžinov and B. I. Zav’jalov,On a multi-dimensional analogue of a theorem of Lindelöf, Sov. Math. Dokl.25 (1982), 51–52.
[H] M. Hervé,Quelques propriétés des applications analytiques d'une boule à m dimensions dans elle-même, J. Math. Pures Appl.42 (1963), 117–147.
[J] F. Jafari,Angular derivatives in polydiscs, Indian J. Math.35 (1993), 197–212.
[J-P] M. Jarnicki and P. Pflug,Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter, Berlin, 1993.
[Ju1] G. Julia,Mémoire sur l’itération des fonctions rationnelles, J. Math. Pures Appl.1 (1918), 47–245.
[Ju2] G. Julia,Extension nouvelle d’un lemme de Schwarz, Acta Math.42 (1920), 349–355.
[K] Yu. V. Khurumov, On Lindelöf’s theorem inC n, Soviet Math. Dokl.28 (1983), 806–809.
[Ko] A. Korányi,Harmonic functions on hermitian hyperbolic spaces, Trans. Amer. Math. Soc.135 (1969), 507–516.
[K-S] A. Korányi and E. M. Stein,Fatou’s theorem for generalized half-planes, Ann. Scuola Norm. Sup. Pisa22 (1968), 107–112.
[K-K] S. Kosbergenov and A. M. Kytmanov,Generalizations of the Schwarz and Riesz-Herglotz formulas in Reinhardt domains (Russian), Izv. Vyssh. Uchebn. Zaved. Mat. no. 10 (1984), 60–63.
[Kr] S. G. Krantz, Invariant metrics and the boundary behavior of holomorphic functions on domains inC n, J. Geom. Anal.1 (1991), 71–97.
[L-V] E. Landau and G. Valiron,A deduction from Schwarz's lemma, J. London Math. Soc.4 (1929), 162–163.
[L] L. Lempert,La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France109 (1981), 427–474.
[N] R. Nevanlinna,Remarques sur le lemme de Schwarz, C. R. Acad. Sci. Paris188 (1929), 1027–1029.
[R] W. Rudin, Function Theory in the Unit Ball ofC n, Springer, Berlin, 1980.
[S] E. M. Stein,The Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972.
[W] J. Wolff,Sur une généralisation d'un théorème de Schwarz, C. R. Acad. Sci. Paris183 (1926), 500–502.
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Abate, M. The Julia-Wolff-Carathéodory theorem in polydisks. J. Anal. Math. 74, 275–306 (1998). https://doi.org/10.1007/BF02819453
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DOI: https://doi.org/10.1007/BF02819453