  A sharp quantitative version of the $L_p-$mixed volume inequality is
established. This is achieved by exploiting an improved Jensen inequality. This
inequality is a generalization of Pinsker-Csisz\'ar-Kullback inequality for the
Tsallis entropy. Finally, a sharp quantitative version of the
$L_p-$Brunn-Minkowski inequality is also proved as a corollary.
