  For a real-valued and essentially bounded arithmetic function $f$, i.e.,
$f(n)\ll_{\varepsilon}\!n^{\varepsilon},\,\forall\varepsilon\!>\!0$, we
\enspace give some optimal links between non-trivial bounds for the sums
$\sum_{h\le H}\sum_{N<n\le 2N}f(n)f(n-h)$, $\sum_{N<x\le 2N} \big| \sum_{x<n\le
x+H}f(n)\big|^2$ and $\sum_{N<n\le 2N} \big| \sum_{0\le |n-x|\le
H}\big(1-{|n-x|\over H}\big)f(n)\big|^2$, with $H=o(N)$ as $N\to\infty$.
