{
  "abstract": "We prove the Koumandos--Ruscheweyh conjecture for every $0<ρ\\leq1$. If $ν(ρ)$ is the unique root in $(0,1]$ of $\\int_0^{(1+ρ)π}t^{μ-1}\\sin(t-ρπ)\\,\\mathrm{d}t=0$, then $(1-z)^ρs_n^μ(z)\\prec((1+z)/(1-z))^ρ$ for every $n\\geq0$, $0<μ\\leqν(ρ)$ and $z\\in\\mathbb{D}$, where $s_n^μ(z)=\\sum_{k=0}^n(μ)_kz^k/k!$. The parameter $ν(ρ)$ is optimal. The proof combines an all-parameter gamma-coefficient comparison with an exact beta integral and a binomial variance estimate, reducing the infinitely many degrees to finitely many continuous interval inequalities. The remaining inequalities are certified by 256-bit ball arithmetic; rational covers, source code and alternative-formula verifiers are provided. The weak positive-real-part conjecture follows as a sharp corollary. We also derive sharp consequences for starlike functions and Gegenbauer polynomial sections: a full-parameter convolution subordination, the optimal starlike order for uniform partial-sum sectors, and the optimal Gegenbauer parameter and sector angle. The necessary bounds and angular sharpness are obtained from explicit kernels and interior scaling limits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02285",
  "author_total": 1,
  "authors": [
    {
      "name": "Yicen Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/00yczmxavtbyk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02285",
  "fulltext_url": null,
  "id": "00yczmxavtbyk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02285",
  "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.CV"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7220458344",
  "pdf_url": "https://arxiv.org/pdf/2610.02285",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/00yczmxavtbyk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.02285",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02285"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Proof of the Koumandos--Ruscheweyh Conjecture",
  "title_text": "A Proof of the Koumandos--Ruscheweyh Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/00yczmxavtbyk",
  "venue": "arXiv",
  "versions": []
}