  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].
