Functional inequalities related to the Rogers-Shephard inequality

For a real-valued nonnegative and log-concave function f defined in R, we introduce a notion of difference function ∆f ; the difference function represents a functional analog on the difference body K + (−K) of a convex body K. We prove a sharp inequality which bounds the integral of ∆f from above, in terms of the integral of f and we characterize equality conditions. The investigation is extended to an analogous notion of difference function for α-concave functions, with α < 0. In this case also, we prove an upper bound for the integral of the α-difference function of f in terms of the integral of f ; the bound is proved to be sharp in the case α = −∞ and in the one dimensional case.