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/

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