{
  "abstract": "We determine the sharp weighted power-mean comparisons for the family $\\{H_a(r)\\}_{r\\in (0,1)}$ with the logarithmic interpretation at $a=0$. In the parameter ranges considered by Barnard, Richards and Tiedeman, we give a complete characterization of all orders $λ,μ\\in\\mathbb{R}$ for which $$A_λ(w;1,1-r)\\leq H_a(r)\\leq A_μ(w;1,1-r)$$ holds for every $r\\in(0,1)$. In particular, our results settle completely their two power-mean conjectures. The two sharp orders are determined by the second-order expansion at $r=0$ and the endpoint matching as $r\\to1$. A weighted Wronskian identity and a sign analysis of its residual establish the global inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28944",
  "author_total": 3,
  "authors": [
    {
      "name": "Qintao Deng"
    },
    {
      "name": "Haoran Peng"
    },
    {
      "name": "Jiatong Zou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5tnka3zh5j6c4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28944",
  "fulltext_url": null,
  "id": "5tnka3zh5j6c4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28944",
  "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": "W7214331949",
  "pdf_url": "https://arxiv.org/pdf/2609.28944",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5tnka3zh5j6c4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28944",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28944"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp power-mean comparisons for Gauss hypergeometric functions",
  "title_text": "Sharp power-mean comparisons for Gauss hypergeometric functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/5tnka3zh5j6c4",
  "venue": "arXiv",
  "versions": []
}