Lower bound for cyclic sums with one-sided maximal averages in denominators

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 $S_n(\mathbf{x},\mathbf{r})={\sum }_{i=1}^n{x}_i\mathrm{/}{a}_i$, 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 $A_{n,\ast }={\inf }_{\mathbf{r}}{\inf }_{\mathbf{x}}{S}_n(\mathbf{x},\mathbf{r})$ as $n\to \mathrm{\infty }$: it is $A_{n,\ast }=e\log n-A+O(1\mathrm{/}\log n)$.