We investigate the monotonicity of the minimal period of the periodic solutions of some quasilinear differential equations and extend results for p=2 due to Chow and Wang, and Chicone, to the case of the p-Laplace operator. Our main result is the monotonicity of the period of optimal functions for a minimization problem related with a fundamental interpolation inequality. In particular we generalize to p≥2 a recent proof of monotonicity due to Benguria, Depassier and~Loss for the same optimality issue and p=2.