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 mean inequality. In a natural randomized setting, we obtain limit theorems and show that a reverse inequality holds with high probability. By its definition, Maclaurin’s inequality is naturally related to U-statistics. More precisely, given $x_1,\dots ,x_n,p\in (0,\mathrm{\infty })$ and $k\in \mathbb{N}$ with $k\le n$, let us define the quantity

$$S_{k,p}^{(n)}={({\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)}^{-1}\sum _{1\le i_1<\cdots <i_k\le n}{x}_{i_1}^p\cdots x_{i_k}^p)}^{1∕(kp)}.$$

Then as a consequence of the classical Maclaurin inequalities, we know that $S_{k_1,p}^{(n)}\ge S_{k_2,p}^{(n)}$ 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+).