An extension of Berwald's inequality and its relation to Zhang's inequality

Abstract

In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function $f:{\mathbb{R}}^n\to [0,\infty )$ and any concave function $h:L\to [0,\infty )$, where $L$ is the epigraph of $-\log \frac{f}{\Vert f{\Vert }_{\infty }}$, then $$p\to {\left(\frac{1}{\Gamma (1+p){\int }_L{e}^{-t}dtdx}{\int }_L{h}^p(x,t)e^{-t}dtdx\right)}^{\frac{1}{p}}$$ is decreasing in $p\in (-1,\infty )$, extending the range of $p$ where the monotonicity is known to hold true. As an application of this extension, we will provide a new proof of a functional form of Zhang's reverse Petty projection inequality, recently obtained in [ABG].