  The purpose of this paper is to study various monotonicity conditions of the
period function $T(c)$ (energy-dependent) for potential systems $\ddot x +
g(x)=0$ with a center at the origin 0. We had before identified a family of new
criteria noted by $(C_n)$ which are sometimes thinner than those previously
known ({\it Period function and characterizations of Isochronous
potentials}\quad arXiv:1109.4611). This fact will be illustrated by examples.
