Functional inequalities related to the Rogers-Shephard inequality

For a real-valued non-negative and log-concave function we introduce a notion of difference function; the difference function represents a functional analog on the difference body of a convex body. We prove a sharp inequality which bounds the integral of the difference function from above in terms of the integral of the function itself, and we characterize equality conditions. The investigation is extended to an analogous notion of difference function for $\alpha$-concave functions, with $\alpha$ smaller than zero. In this case also, we prove an upper bound for the integral of the $\alpha$-difference function of a function in terms of the integral of the function itself; the bound is proved to be sharp when $\alpha$ is equal to minus infinity and in the one dimensional case.