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

Abstract

Let X be a complex Banach manifold. A holomorphic function f : X → C is called a normal function if the family F f = { f ◦φ : φ ∈ O( , X)} forms a normal family in the sense of Montel (here O( , 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 {xn} be a sequence of points in D which tends to a boundary point ξ ∈ ∂ D such that limn→∞ f (xn) = L for some L ∈ C. Sufficient conditions on a sequence {xn} 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. Mathematics Subject Classification (2000): 32A18 (primary).