The Maclaurin inequality through the probabilistic lens

Abstract

In this paper we take a probabilistic look at Maclaurin's inequality, which is a refinement of the classical AM-GM inequality. In a natural randomized setting, we obtain limit theorems and show that a reverse inequality holds with high probability. The form of Maclaurin's inequality naturally relates it to U-statistics. More precisely, given $x_1,\dots ,x_n,p\in (0,\infty )$ and $k\in \mathbb{N}$ with $k\le n$, let us define the quantity \[ S_{k, p}^{(n)} = \Big( \tbinom{n}{k}^{-1} \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1}^p \cdots x_{i_k}^p \Big)^{1/(k p)}.\] Then as a consequence of the classical Maclaurin inequalities, we know that $S_{k_1}^{(n)}\ge S_{k_2}^{(n)}$ for $k_1<k_2$. In the present article we consider the ratio \[ \mathcal{R}_{k_1, k_2, p}^{(n)} := \frac{S_{k_2, p}^{(n)}}{S_{k_1, p}^{(n)}}, \] evaluated at a random vector $(X_1,\dots ,X_n)$ sampled either from the normalized surface measure on the ${\ell }_p^n$-sphere or from a distribution generalizing both the uniform distribution on the ${\ell }_p^n$-ball and the cone measure on the ${\ell }_p^n$-sphere; by the Maclaurin inequality, we always have ${\mathcal{R}}_{k_1,k_2,p}^{(n)}\le 1$. We derive central limit theorems for ${\mathcal{R}}_{k_1,k_2,p}^{(n)}$ and ${\mathcal{R}}_{k_1,n,p}^{(n)}$ as well as Berry--Esseen bounds and a moderate deviations principle for ${\mathcal{R}}_{k_1,n,p}^{(n)}$, keeping $k_1$, $k_2$ fixed, in order to quantify the set of points where ${\mathcal{R}}_{k_1,k_2,p}^{(n)}>c$ for $c\in (0,1)$, i.e., where the Maclaurin inequality is reversed up to a factor. The present aricle partly generalizes results concerning the AM-GM inequality obtained by Kabluchko, Prochno, and Vysotsky (2020), Thäle (2021), and Kaufmann and Thäle (2023+).

30 pages