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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 $p\ge2$ a recent proof of
monotonicity due to Benguria, Depassier and~Loss for the same optimality issue
and $p=2$.
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