A Maclaurin type inequality

Abstract

The classical Maclaurin inequality asserts that the elementary symmetric means $$s_k(y)=\frac{1}{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}\sum _{1\le i_1<\cdots <i_k\le n}{y}_{i_1}\dots y_{i_k}$$ obey the inequality $s_{\ell }(y{)}^{1/\ell }\le s_k(y{)}^{1/k}$ whenever $1\le k\le \ell \le n$ 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.

12 pages, no figures. Minor typo corrections