2024 The Maclaurin inequality through the probabilistic lens
Lorenz Frühwirth, Michael Juhos, Joscha Prochno
Electron. J. Probab. 29: 1-33 (2024). DOI: 10.1214/24-EJP1165

## 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 ${x}_{1},\dots ,{x}_{n},p\in \left(0,\mathrm{\infty }\right)$ and $k\in \mathbb{N}$ with $k\le n$, let us define the quantity

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

Then as a consequence of the classical Maclaurin inequalities, we know that ${S}_{{k}_{1},p}^{\left(n\right)}\ge {S}_{{k}_{2},p}^{\left(n\right)}$ for ${k}_{1}<{k}_{2}$. In the present article we consider the ratio

${\mathcal{R}}_{{k}_{1},{k}_{2},p}^{\left(n\right)}:=\frac{{S}_{{k}_{2},p}^{\left(n\right)}}{{S}_{{k}_{1},p}^{\left(n\right)}},$

evaluated at a random vector $\left({X}_{1},\dots ,{X}_{n}\right)$ 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 ${\mathcal{R}}_{{k}_{1},{k}_{2},p}^{\left(n\right)}\le 1$. We derive central limit theorems for ${\mathcal{R}}_{{k}_{1},{k}_{2},p}^{\left(n\right)}$ and ${\mathcal{R}}_{{k}_{1},n,p}^{\left(n\right)}$ as well as Berry–Esseen bounds and a moderate deviations principle for ${\mathcal{R}}_{{k}_{1},n,p}^{\left(n\right)}$, keeping ${k}_{1}$, ${k}_{2}$ fixed, in order to quantify the set of points where ${\mathcal{R}}_{{k}_{1},{k}_{2},p}^{\left(n\right)}>c$ for $c\in \left(0,1\right)$, 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+).

## Funding Statement

LF was supported by the Austrian Science Fund (FWF), projects P-32405 and P-35322. JP is supported by the German Research Foundation (DFG) under project 516672205 and by the Austrian Science Fund (FWF) under project P-32405.

## Citation

Lorenz Frühwirth. Michael Juhos. Joscha Prochno. "The Maclaurin inequality through the probabilistic lens." Electron. J. Probab. 29 1 - 33, 2024. https://doi.org/10.1214/24-EJP1165

## Information

Received: 20 December 2023; Accepted: 27 June 2024; Published: 2024
First available in Project Euclid: 30 July 2024

arXiv: 2312.12134
Digital Object Identifier: 10.1214/24-EJP1165

Subjects:
Primary: 60D05 , 60F99
Secondary: 46N30

Keywords: functional analysis , MacLaurin inequality , Probability

JOURNAL ARTICLE
33 PAGES

Vol.29 • 2024