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

Abstract

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 f is finite, then f cannot be hypercyclic for the derivative operator; and, if a positive function ϕ having infinite exponential growth is fixed, then there exist zero-free hypercyclic functions which are controlled by ϕ along a sequence of radii tending to infinity.