So, we managed to prove the following inequality:
Let $f$ be a *decreasing* *convex* function on $\mathbb{R}_{\geq 0}$, let $p$ be a positive integer and $r\geq 0$.
Then
\begin{align}
\frac{1}{p} \sum_{\ell=1}^{p} f\left(\frac{\ell+r}{p} \right) \leq \frac{1}{p+1} \sum_{\ell=1}^{p+1} f\left( \frac{\ell+r}{p+1} \right).
\end{align}

**Does this inequality have a name?** Is it known?

It is related to this question over at math.stackexchange.