The period functions' higher order derivatives

We prove a formula for the $n$-th derivative of the period function $T$ in a period annulus of a planar differential system. For $n=1$, we obtain Freire, Gasull and Guillamon formula for the period’s first derivative \cite{FGG}. We apply such a result to hamiltonian systems with separable variables and other systems. We give some sufficient conditions for the period function of conservative second order O.D.E.'s to be convex.