On normal and non-normal holomorphic functions on complex Banach manifolds
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 1-15.

Let X be a complex Banach manifold. A holomorphic function f:X is called a normal function if the family f ={fφ:φ𝒪(Δ,X)} forms a normal family in the sense of Montel (here 𝒪(Δ,X) denotes the set of all holomorphic maps from the complex unit disc into X). Characterizations of normal functions are presented. A sufficient condition for the sum of a normal function and non-normal function to be non-normal is given. Criteria for a holomorphic function to be non-normal are obtained. These results are used to draw one interesting conclusion on the boundary behavior of normal holomorphic functions in a convex bounded domain D in a complex Banach space V. Let {x n } be a sequence of points in D which tends to a boundary point ξD such that lim n f(x n )=L for some L ¯. Sufficient conditions on a sequence {x n } of points in D and a normal holomorphic function f are given for f to have the admissible limit value L, thus extending the result obtained by Bagemihl and Seidel.

Classification : 32A18
Dovbush, Peter 1

1 Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova, 5 Academy Street, Kishinev, MD-2028, Republic of Moldova
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Dovbush, Peter. On normal and non-normal holomorphic functions on complex Banach manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 1-15. http://www.numdam.org/item/ASNSP_2009_5_8_1_1_0/

[1] F. Bagemihl and W. Seidel, Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 280 (1960). | MR | Zbl

[2] D. M. Campbell and G. Wickes, Characterizations of normal meromorphic functions, In: “Complex Analysis”, Laine, J. et al. (eds.), Joensuu 1978, Lect. Notes Math., Vol. 747, Springer, Berlin-Heidelberg-New York, 1979, 55–72. | Zbl

[3] S. Dineen, “The Schwarz Lemma”, Oxford Mathematical Monographs, Oxford, 1989. | MR | Zbl

[4] C. J. Earle, L. A. Harris, J. H. Hubbard and S. Mitra, Schwarz’s lemma and the Kobayashi and Carathéodory metrics on complex Banach manifolds, In: “Kleinian Groups and Hyperbolic 3-Manifolds”, Cambridge Univ. Press, Cambridge, Lond. Math. Soc. Lec. Notes, Vol. 299, 2003, 363–384. http://www.ms.uky.edu/ larry/paper.dir/minsky.ps. | Zbl

[5] T. Franzoni and E. Vesentini, “Holomorphic Maps and Invariant Distances”, North-Holland Mathematical Studies 40, North-Holland Publishing, Amsterdam, 1980. | MR | Zbl

[6] V. I. Gavrilov, Boundary properties of functions meromorphic in the unit disc, Dokl. Akad. Nauk SSSR 151 (1963), 19-22 (in Russian). | MR

[7] P. Gauthier, A criterion of normalcy, Nagoya Math. J. 32 (1968), 272–282. | MR | Zbl

[8] K. T. Hahn, Non-tangential limit theorems for normal mappings, Pacific J. Math. 135 (1988), 57–64. | MR | Zbl

[9] M. H. Kwack, “Families of Normal Maps in Several Variables and Classical Theorems in Complex Analysis”, Lecture Notes Series, Vol. 33, Res. Inst. Math., Global Analysis Res. Center, Seoul, Korea, 1996. | MR | Zbl

[10] P. Lappan, Non-normal sums and prodacts of unbounded normal functions, Michigan Math. J. 8 (1961), 187–192. | MR | Zbl

[11] P. Lappan, Normal families and normal functions: results and techniques, In: “Function Spaces and Complex Analysis”, Joensuu 1997, Univ. Joensuu, Department of Mathematics Rep. Ser. 2 (1997), 63–78. | MR | Zbl

[12] O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47–65. | MR | Zbl

[13] A. J. Lohwater, The boundary behavior of analytic functions, In: “Current Problems in Mathematics, Fundamental Directions”, Vol. 10, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1973, 99–259 (in Russian). | Zbl

[14] K. Noshiro, Contributions to the theory of meromorphic functions in the unit circle, J. Fac. Sci. Hokkaido Imp. Univ., Ser. I (1938), 149–159. | JFM

[15] J. L. Shchiff, “Normal Families”, Springer, New York, 1993. | MR

[16] K. Yosida, On a class of meromorphic functions Proc. Phys.-Math. Soc. Japan, ser. 1, 3 (1934), 227–235. | JFM

[17] M. G. Zaidenberg, Schottky-Landau growth estimates for s-normal families of holomorphic mappings, Math. Ann. 293 (1992), 123–141. | EuDML | MR

[18] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230. | MR | Zbl