Derivative and factorization of holomorphic functions

For a holomorphic function f of bounded type on a complex Banach space E, we show that its derivative df:E→E∗ takes bounded sets into certain families of sets if and only if f may be factored in the form f=g○S, where S is in some associated operator ideal, and g is a holomorphic function of bounded type. We also prove that the multilinear and polynomial mappings factor in an analogous way if and only if they are “K-bounded.”