Improved LpLp-mixed volume inequality for convex bodies

A sharp quantitative version of the LpLp-mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker–Csiszár–Kullback inequality for the Tsallis entropy. Finally, a sharp quantitative version of the LpLp-Brunn–Minkowski inequality is also proved as a corollary.