  Let $\mathbf{x}=(x_1,\dots,x_n)$ be an $n$-tuple of positive real numbers and
the sequence $(x_i)_{i\in\mathbb{Z}}$ be its $n$-periodic extension. Given an
$n$-tuple $\mathbf{r}=(r_1,\dots,r_n)$ of positive integers, let $a_i$ be the
arithmetic mean of $x_{i+1},\dots,x_{i+r_i}$. We form the cyclic sums
$S_n(\mathbf{x},\mathbf{r})=\sum_{i=1}^n x_i/a_{i}$, following the pattern of
the long studied Shapiro sums, which correspond to all $r_i=2$, and more
general Diananda sums, where all $r_i$ are equal. We find the asymptotics of
the $\mathbf{r}$-independent lower bounds
$A_{n,*}=\inf_{\mathbf{r}}\inf_{\mathbf{x}} S_n(\mathbf{x},\mathbf{r})$ as
$n\to\infty$: it is $A_{n,*}=e\log n - A+O(1/\log n)$.
