Optimal growth of entire functions frequently hypercyclic for the differentiation operator ✩

We solve a problem posed by Bonilla and Grosse-Erdmann (2007) [7] by constructing an entire function f which is frequently hypercyclic with respect to the differentiation operator, and satisfies Mf (r) cer r−1/4, where c > 0 may be chosen arbitrarily small. This growth rate is sharp. We also obtain optimal results for minimal growth in terms of average Lp-norms. Among other tools, the proof uses the Rudin– Shapiro polynomials and heat kernel estimates. © 2012 Elsevier Inc. All rights reserved