Let be a complex Banach manifold. A holomorphic function is called a normal function if the family forms a normal family in the sense of Montel (here denotes the set of all holomorphic maps from the complex unit disc into ). 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 in a complex Banach space Let be a sequence of points in which tends to a boundary point such that for some Sufficient conditions on a sequence of points in and a normal holomorphic function are given for to have the admissible limit value thus extending the result obtained by Bagemihl and Seidel.
@article{ASNSP_2009_5_8_1_1_0, author = {Dovbush, Peter}, title = {On normal and non-normal holomorphic functions on complex {Banach} manifolds}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {1--15}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 8}, number = {1}, year = {2009}, mrnumber = {2512198}, zbl = {1183.32004}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2009_5_8_1_1_0/} }
TY - JOUR AU - Dovbush, Peter TI - On normal and non-normal holomorphic functions on complex Banach manifolds JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2009 SP - 1 EP - 15 VL - 8 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2009_5_8_1_1_0/ LA - en ID - ASNSP_2009_5_8_1_1_0 ER -
%0 Journal Article %A Dovbush, Peter %T On normal and non-normal holomorphic functions on complex Banach manifolds %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2009 %P 1-15 %V 8 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2009_5_8_1_1_0/ %G en %F ASNSP_2009_5_8_1_1_0
Dovbush, Peter. On normal and non-normal holomorphic functions on complex Banach manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 1, pp. 1-15. http://archive.numdam.org/item/ASNSP_2009_5_8_1_1_0/
[1] Sequential and continuous limits of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 280 (1960). | MR | Zbl
and ,[2] 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
and ,[3] “The Schwarz Lemma”, Oxford Mathematical Monographs, Oxford, 1989. | MR | Zbl
,[4] 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
, , and ,[5] “Holomorphic Maps and Invariant Distances”, North-Holland Mathematical Studies 40, North-Holland Publishing, Amsterdam, 1980. | MR | Zbl
and ,[6] Boundary properties of functions meromorphic in the unit disc, Dokl. Akad. Nauk SSSR 151 (1963), 19-22 (in Russian). | MR
,[7] A criterion of normalcy, Nagoya Math. J. 32 (1968), 272–282. | MR | Zbl
,[8] Non-tangential limit theorems for normal mappings, Pacific J. Math. 135 (1988), 57–64. | MR | Zbl
,[9] “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] Non-normal sums and prodacts of unbounded normal functions, Michigan Math. J. 8 (1961), 187–192. | MR | Zbl
,[11] 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] Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47–65. | MR | Zbl
and ,[13] 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] Contributions to the theory of meromorphic functions in the unit circle, J. Fac. Sci. Hokkaido Imp. Univ., Ser. I (1938), 149–159. | JFM
,[15] “Normal Families”, Springer, New York, 1993. | MR
,[16] On a class of meromorphic functions Proc. Phys.-Math. Soc. Japan, ser. 1, 3 (1934), 227–235. | JFM
,[17] Schottky-Landau growth estimates for s-normal families of holomorphic mappings, Math. Ann. 293 (1992), 123–141. | EuDML | MR
,[18] Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230. | MR | Zbl
,