{
  "abstract": "We prove McLeod's (1959) conjecture that $(h_{\\ell+r}/h_\\ell)^{1/r}$ is subadditive for all $\\ell\\ge0$ and $r\\ge1$, where $h_k$ is the complete homogeneous symmetric polynomial of degree $k$ (J. B. McLeod, \"On four inequalities in symmetric functions\", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio $h_k/h_{k-1}$ is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of $r$ consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07302",
  "author_total": 1,
  "authors": [
    {
      "name": "Suvrit Sra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4jd2q1d1mmcn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07302",
  "fulltext_url": null,
  "id": "c4jd2q1d1mmcn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07302",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7220952324",
  "pdf_url": "https://arxiv.org/pdf/2610.07302",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4jd2q1d1mmcn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07302",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07302"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios",
  "title_text": "A Proof of McLeod's 1959 Conjecture on Complete Homogeneous Symmetric Ratios",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/c4jd2q1d1mmcn",
  "venue": "arXiv",
  "versions": []
}