On normal and non-normal holomorphic functions on complex Banach manifolds

Dovbush, Peter

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 $f:X\to ℂ$ is called a normal function if the family ${ℱ}_f=\{f\circ \phi :\phi \in 𝒪(\Delta ,X)\}$ forms a normal family in the sense of Montel (here $𝒪(\Delta ,X)$ 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 $\left\{x_n\right\}$ 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.

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