Quantitative improvements of functional inequalities under concavity properties

A classical result of Hensley provides a sharp lower bound for the functional $\int_\mathbb{R} t^2f$, 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 $s\geq 0$. Second, we present a generalization of Barthe and Koldobsky's result for functionals of the form $\int_\mathbb{R} Nf\,\mathrm{d}\mu$, where $N$ is a convex, even function and $\mu$ 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.