Optimal growth of frequently hypercyclic entire functions

We solve a problem posed by A. Bonilla and K.-G. Grosse-Erdmann by constructing an entire function $f$ that is frequently hypercyclic with respect to the differentiation operator, and satisfies $M_f(r)\leq\displaystyle ce^r r^{-1/4}$, where $c>0$ be chosen arbirarily small. The obtained growth rate is sharp. We also obtain optimal results for the growth when measured in terms of average $L^p$-norms. Among other things, the proof applies Rudin-Shapiro polynomials and heat kernel estimates.