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 to Chicone, to the case of the p-Laplace operator. Our main result is the monotonicity of the period for positive solutions of a nonlinear Euler-Lagrange equation for a minimization problem related with a fundamental interpolation inequality. In particular we generalize to p greater than 2 recent results of Benguria, Depassier and Loss.