A new arrangement inequality (revised and extended version)

Abstract

We study the cyclic inequality \[ \sum_{i=1}^n x_i\,\sum_{i=1}^n x_i^a x_{i+1}^b\;\le\;\Big(\sum_{i=1}^n x_i^{(1+a+b)/2}\Big)^2, \] where $x_1,\dots,x_n\ge 0$ and indices are taken modulo $n$. Let $N(a,b)$ be the largest $n$ for which the inequality holds for all nonnegative $x_i$. We show that $N(a,b)$ is bounded above by an explicit quantity $N^*(a,b)$ obtained from the second variation at the constant vector, and we conjecture that $N(a,b)=N^*(a,b)$. For $a=b=1$ we prove that the inequality holds if and only if $n\le 8$, which settles the case left open in the original version of this paper. For $n=4$ the inequality holds whenever $1,a,b$ are the sides of a triangle, with an explicit remainder term. We show that $N$ is monotone along rays emanating from the line $a+b=1$; this corrects and extends an earlier monotonicity statement. Finally, any $n$ nonnegative numbers can be arranged so that the inequality holds; for $n\neq 6$ this is achieved by one of $n-1$ prescribed arrangements, which yields a volume bound for every such $n$.