On universal entire functions with zero-free derivatives

We prove in this note a generalization of a theorem due to G. Herzog on zero-free universal entire functions. Specifically, it is shown that, if a nonnegative integer q and a nonconstant entire function Phi of subexponential type are given, then there is a residual set in the class of entire functions with zero-free derivatives of orders q and q + 1, such that every member of that set is universal with respect to Phi(D), where D is the differentiation operator.