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

TL;DR

This paper establishes the asymptotic behavior of the minimal cyclic sums involving one-sided averages in denominators, generalizing previous sums and providing precise growth rates as the number of terms increases.

Contribution

It derives the asymptotic formula for the lower bounds of cyclic sums with variable averaging lengths, extending known results for specific sum types.

Findings

Asymptotic lower bounds grow as e log n minus a constant

Generalizes Shapiro and Diananda sums to variable r_i sums

Provides precise asymptotic behavior as n approaches infinity

Abstract

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 $A_{n,*}=e\log n - A+O(1/\log n)$.