Quantitative improvements of functional inequalities under concavity properties

A classical result of Hensley provides a sharp lower bound for the functional ∫R t²f, where f is a non-negative, even log-concave function. In the context of studying the minimal slabs of the unit cube, Barthe and Koldobsky established a quantitative improvement of Hensley's bound. In this work, we complement their result in several directions. First, we prove the corresponding upper bound inequality for s-concave functions with s≥ 0. Second, we present a generalization of Barthe and Koldobsky's result for functionals of the form ∫R Nf\,dμ, where N is a convex, even function and $μ$ belongs to a suitable class of positive Borel measures. As a consequence of the employed methods, we obtain quantitative refinements of classical inequalities for p-norms and for the entropy of log-concave functions. Finally, we discuss both geometric consequences and probabilistic interpretations of our results.