T. Tao. (2023)cite arxiv:2310.05328Comment: 11 pages, no figures.
Abstract
The classical Maclaurin inequality asserts that the elementary symmetric
means $$ s_k(y) = 1nk \sum_1 i_1 < < i_k n
y_i_1 y_i_k$$ obey the inequality $s_\ell(y)^1/\ell łeq
s_k(y)^1/k$ whenever $1 k n$ and $y = (y_1,\dots,y_n)$
consists of non-negative reals. We establish a variant $$
|s_\ell(y)|^1\ell \ell^1/2k^1/2 \max
(|s_k(y)|^1k, |s_k+1(y)|^1k+1)$$ of this inequality in
which the $y_i$ are permitted to be negative. In this regime the inequality is
sharp up to constants. Such an inequality was previously known without the
$k^1/2$ factor in the denominator.
%0 Generic
%1 tao2023maclaurin
%A Tao, Terence
%D 2023
%K analysis maclaurin mathematics
%T A Maclaurin type inequality
%U http://arxiv.org/abs/2310.05328
%X The classical Maclaurin inequality asserts that the elementary symmetric
means $$ s_k(y) = 1nk \sum_1 i_1 < < i_k n
y_i_1 y_i_k$$ obey the inequality $s_\ell(y)^1/\ell łeq
s_k(y)^1/k$ whenever $1 k n$ and $y = (y_1,\dots,y_n)$
consists of non-negative reals. We establish a variant $$
|s_\ell(y)|^1\ell \ell^1/2k^1/2 \max
(|s_k(y)|^1k, |s_k+1(y)|^1k+1)$$ of this inequality in
which the $y_i$ are permitted to be negative. In this regime the inequality is
sharp up to constants. Such an inequality was previously known without the
$k^1/2$ factor in the denominator.
@misc{tao2023maclaurin,
abstract = {The classical Maclaurin inequality asserts that the elementary symmetric
means $$ s_k(y) = \frac{1}{\binom{n}{k}} \sum_{1 \leq i_1 < \dots < i_k \leq n}
y_{i_1} \dots y_{i_k}$$ obey the inequality $s_\ell(y)^{1/\ell} \leq
s_k(y)^{1/k}$ whenever $1 \leq k \leq \ell \leq n$ and $y = (y_1,\dots,y_n)$
consists of non-negative reals. We establish a variant $$
|s_\ell(y)|^{\frac{1}{\ell}} \ll \frac{\ell^{1/2}}{k^{1/2}} \max
(|s_k(y)|^{\frac{1}{k}}, |s_{k+1}(y)|^{\frac{1}{k+1}})$$ of this inequality in
which the $y_i$ are permitted to be negative. In this regime the inequality is
sharp up to constants. Such an inequality was previously known without the
$k^{1/2}$ factor in the denominator.},
added-at = {2023-10-21T09:43:00.000+0200},
author = {Tao, Terence},
biburl = {https://www.bibsonomy.org/bibtex/2a0d8e7883ae2364e9c26739b01de788c/tabularii},
description = {[2310.05328] A Maclaurin type inequality},
interhash = {b40f1e9fad5747dfe1a45a218f4e515d},
intrahash = {a0d8e7883ae2364e9c26739b01de788c},
keywords = {analysis maclaurin mathematics},
note = {cite arxiv:2310.05328Comment: 11 pages, no figures},
timestamp = {2023-10-21T09:43:00.000+0200},
title = {A Maclaurin type inequality},
url = {http://arxiv.org/abs/2310.05328},
year = 2023
}