{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-maclaurin-type-inequality","title":"A Maclaurin type inequality","arxiv_id":"2310.05328","date":"2023-10-09","proceeding":null,"authors":["Terence Tao"],"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.","url_abs":"https://arxiv.org/abs/2310.05328v3","url_pdf":"https://arxiv.org/pdf/2310.05328v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-maclaurin-type-inequality","repo_url":"https://github.com/teorth/symmetric_project","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2310.05328","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}