{
  "abstract": "The univalent Bloch-Landau constant $\\mathcal{U}$ is the largest number such that, for every univalent function $f$ on the unit disk, $f(\\mathbb{D})$ contains a disk of radius $\\mathcal{U} |f'(0)|$. We prove $0.612242\\le\\mathcal{U}<0.6562681$. The previous lower bound, $0.57088586$, came from growth estimates, which control disks centered at $f(0)$. Our improvement rests on a different mechanism. After Landau's normalization, a $2\\times2$ compression of the Grunsky matrix constrains every omitted value through the same two coefficients, and choosing the center of the disk from these coefficients exploits this shared information. A short analytic proof based on this idea gives $\\mathcal{U}\\ge0.592$; it has been formalized in Lean, with the classical Grunsky and Fekete-Szegő inequalities as the only external inputs. Retaining complex quadratic forms of the matrix gives $\\mathcal{U}\\ge0.612242$ through a finite interval certificate, and this bound holds for every normalized univalent function with $a_2=0$ and $|a_3|\\le1/3$. We also show that no argument based on disks centered at $f(0)$ can give a bound above $0.6$. For the upper bound we construct an explicit domain, bounded by Goodman's first two slit generations, thirty analytic arcs and a circle. We certify $\\mathcal{U}<0.6562681$ with a capacity estimate for almost critical arcs, improving the best published bound $0.6563937$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34580",
  "author_total": 1,
  "authors": [
    {
      "name": "Frank Wikström"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7120dkq7mn2pg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34580",
  "fulltext_url": null,
  "id": "7120dkq7mn2pg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34580",
  "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": "W7223915984",
  "pdf_url": "https://arxiv.org/pdf/2610.34580",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7120dkq7mn2pg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34580",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34580"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Bounds for the univalent Bloch-Landau constant",
  "title_text": "Bounds for the univalent Bloch-Landau constant",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7120dkq7mn2pg",
  "venue": "arXiv",
  "versions": []
}