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

Let X be a complex Banach manifold. A holomorphic function f : X -> C is called a normal function if the family F-f = {f o phi : phi is an element of 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 {x(n)} be a sequence of points in D which tends to a boundary point xi is an element of partial derivative D such that lim(n ->infinity) f(x(n)) = L for some L is an element of (C) over bar. 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.