  We construct a dynamically convex contact form on the three-sphere whose
systolic ratio is arbitrarily close to 2. This example is related to a
conjecture of Viterbo, whose validity would imply that the systolic ratio of a
convex contact form does not exceed 1. We also construct a sequence of tight
contact forms $\alpha_n$, $n\geq 2$, with systolic ratio arbitrarily close to
$n$ and suitable bounds on the mean rotation number of all the closed orbits of
the induced Reeb flow.
