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

In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function \(f:\mathbb R^n\rightarrow[0,\infty)\) and any concave function \(h:L\rightarrow\mathbb [0,\infty)\), 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].