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, p. 1-15

Let $X$ be a complex Banach manifold. A holomorphic function $f:X\to ℂ$ is called a normal function if the family ${ℱ}_{f}=\left\{f\circ \phi :\phi \in 𝒪\left(\Delta ,X\right)\right\}$ forms a normal family in the sense of Montel (here $𝒪\left(\Delta ,X\right)$ 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 $\left\{{x}_{n}\right\}$ be a sequence of points in $D$ which tends to a boundary point $\xi \in \partial D$ such that ${lim}_{n\to \infty }f\left({x}_{n}\right)=L$ for some $L\in \overline{ℂ}.$ Sufficient conditions on a sequence $\left\{{x}_{n}\right\}$ 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
@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},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {1},
year = {2009},
pages = {1-15},
zbl = {1183.32004},
mrnumber = {2512198},
language = {en},
url = {http://www.numdam.org/item/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://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 121488 | Zbl 0095.05801

[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 0497.30026

[3] S. Dineen, “The Schwarz Lemma”, Oxford Mathematical Monographs, Oxford, 1989. | MR 1033739 | Zbl 0708.46046

[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 1047.32008

[5] T. Franzoni and E. Vesentini, “Holomorphic Maps and Invariant Distances”, North-Holland Mathematical Studies 40, North-Holland Publishing, Amsterdam, 1980. | MR 563329 | Zbl 0447.46040

[6] V. I. Gavrilov, Boundary properties of functions meromorphic in the unit disc, Dokl. Akad. Nauk SSSR 151 (1963), 19-22 (in Russian). | MR 152658

[7] P. Gauthier, A criterion of normalcy, Nagoya Math. J. 32 (1968), 272–282. | MR 230891 | Zbl 0157.39802

[8] K. T. Hahn, Non-tangential limit theorems for normal mappings, Pacific J. Math. 135 (1988), 57–64. | MR 965684 | Zbl 0618.32004

[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 1406567 | Zbl 0863.32004

[10] P. Lappan, Non-normal sums and prodacts of unbounded normal functions, Michigan Math. J. 8 (1961), 187–192. | MR 131554 | Zbl 0133.03603

[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 1712274 | Zbl 1129.30314

[12] O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47–65. | MR 87746 | Zbl 0077.07702

[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 0283.30032

[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 65.0334.01

[15] J. L. Shchiff, “Normal Families”, Springer, New York, 1993. | MR 1211641

[16] K. Yosida, On a class of meromorphic functions Proc. Phys.-Math. Soc. Japan, ser. 1, 3 (1934), 227–235. | JFM 60.0266.04

[17] M. G. Zaidenberg, Schottky-Landau growth estimates for s-normal families of holomorphic mappings, Math. Ann. 293 (1992), 123–141. | MR 1162678

[18] L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230. | MR 1624862 | Zbl 1037.30021