On the growth of zero-free MacLane-universal entire functions

We show that exponential growth is the critical discrete rate of growth for zero-free entire functions which are universal in the sense of MacLane. Specifically, it is proved that, if the lower exponential growth order of a zero-free entire function ff is finite, then ff cannot be hypercyclic for the derivative operator; and, if a positive function φφ having infinite exponential growth is fixed, then there exist zero-free hypercyclic functions which are controlled by φφ along a sequence of radii tending to infinity.