  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.
