On universal entire functions with zero-free derivatives (Q1816482)

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 superexponential 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.