{
  "abstract": "Let \\begin{equation*} f_p(x)=\\frac{\\mathcal{K}(\\sqrt{x})}{p-\\ln\\sqrt{1-x}}, \\end{equation*} where $\\mathcal{K}$ denotes the complete elliptic integral of the first kind, and set $g_p=1/f_p$. Tian and Yang conjectured that, at $p=\\ln4$, both $-f_{\\ln4}^{\\prime\\prime\\prime}$ and $g_{\\ln4}$ are absolutely monotonic on $(0,1)$. Writing \\begin{equation*} \\frac{2}πf_{\\ln4}(x)=\\sum_{n=0}^{\\infty}a_n(\\ln4)x^n,\\quad \\fracπ{2}g_{\\ln4}(x)=\\sum_{n=0}^{\\infty}b_n(\\ln4)x^n, \\end{equation*} we prove \\begin{equation*} a_n(\\ln4)<0\\quad(n\\geq3),\\quad b_n(\\ln4)>0\\quad(n\\geq0), \\end{equation*} thereby settling both conjectures. The proof of the first inequality is based on the representation \\begin{equation*} a_n(\\ln4)=\\frac{(-1)^nC}{15^{n+1}}-M_n, \\end{equation*} where $C>0$ and $(M_n)$ is a positive moment sequence. Its strict log-convexity, together with estimates for the initial coefficients, determines the sign of every $a_n$. The second inequality follows from a factorization of the quadratic truncation of the normalized Taylor series and a reciprocal-series argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32366",
  "author_total": 3,
  "authors": [
    {
      "name": "Qintao Deng"
    },
    {
      "name": "Yunjia Kou"
    },
    {
      "name": "Fulin Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ztdb7nq1ad7a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32366",
  "fulltext_url": null,
  "id": "5ztdb7nq1ad7a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32366",
  "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",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214766340",
  "pdf_url": "https://arxiv.org/pdf/2609.32366",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ztdb7nq1ad7a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32366",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32366"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Endpoint Absolute Monotonicity for the Complete Elliptic Integral of the First Kind",
  "title_text": "Endpoint Absolute Monotonicity for the Complete Elliptic Integral of the First Kind",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5ztdb7nq1ad7a",
  "venue": "arXiv",
  "versions": []
}