Abstract

Let X be a complex Banach manifold. A holomorphic function f : X → C is called a Bloch function if the family Ff = {f o ρ – f(ρ(0)) : ρ : Δ → X is holomorphic} , Δ = {z ∈ C : |z| < 1}, is a normal family in the sense of Montel. In this paper Bloch functions on complex Banach manifolds are studied. The main result shows that many of the equivalent definitions of Bloch functions on the unit disk are also equivalent in the general setting.