  We give a sharp lower bound for the number of geometrically distinct
contractible periodic orbits of dynamically convex Reeb flows on
prequantizations of symplectic manifolds that are not aspherical. Several
consequences of this result are obtained, like a new proof that every bumpy
Finsler metric on $S^n$ carries at least two prime closed geodesics,
multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically
convex contact forms with finitely many geometrically distinct contractible
closed orbits and precise estimates of the number of even periodic orbits of
perfect contact forms. We also slightly relax the hypothesis of dynamical
convexity. A fundamental ingredient in our proofs is the common index jump
theorem due to Y. Long and C. Zhu.
