{
  "abstract": "For $-1<α\\leq1/2$, let $J_α$ be the Bessel function of the first kind and let $j_{α,2}$ be its second positive zero. Define $β(α)<α+1$ by $$ \\int_0^{j_{α,2}}u^{-β(α)}J_α(u)\\,du=0. $$ We prove that $β''(α)>0$ for $-1<α\\leq1/2$, including the left second derivative at $α=1/2$. The continuous extension $β(-1)=0$ is strictly convex on $[-1,1/2]$, strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10209",
  "author_total": 2,
  "authors": [
    {
      "name": "K. Castillo"
    },
    {
      "name": "S. Yakubovich"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2jh8xayyshm54.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10209",
  "fulltext_url": null,
  "id": "2jh8xayyshm54",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10209",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7212171987",
  "pdf_url": "https://arxiv.org/pdf/2609.10209",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2jh8xayyshm54.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10209",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10209"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A proof of Askey's convexity conjecture",
  "title_text": "A proof of Askey's convexity conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/2jh8xayyshm54",
  "venue": "arXiv",
  "versions": []
}