{
  "abstract": "Let $Σ$ denote the class of functions $f(z) = z + b_0 + b_1 z^{-1} + \\cdots$ that are analytic and univalent in the exterior unit disk $Δ^* = \\{z \\in \\mathbb{C} : |z| > 1\\}$. In 1962, Ch. Pommerenke proved that if $F$ and $G$ are convex functions in $Σ$, then every convex linear combination $H = λF + (1-λ)G$ ($0 < λ< 1$) remains univalent and belongs to $Σ$. In Hayman's problem collection (Research Problems in Function Theory, Problem 6.10), Pommerenke raised the question of whether $H$ is necessarily also a convex function. We resolve this question in the negative by constructing an explicit counterexample in $Σ$ with parameters in $\\mathbb{Q}(i)$. The argument is self-contained and has been formally certified in the Lean 4 proof assistant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04279",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuankai Guo"
    },
    {
      "name": "Xiaozhe Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0q192qwypmnmd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04279",
  "fulltext_url": null,
  "id": "0q192qwypmnmd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04279",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7204632840",
  "pdf_url": "https://arxiv.org/pdf/2609.04279",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0q192qwypmnmd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04279",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04279"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Counterexample to a Problem of Pommerenke on Convex Functions in the Class $Σ$",
  "title_text": "A Counterexample to a Problem of Pommerenke on Convex Functions in the Class Σ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/0q192qwypmnmd",
  "venue": "arXiv",
  "versions": []
}