%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 }