{
  "abstract": "We disprove a conjecture of Liu, Luo, and Luo (2020) concerning the Landau radius for bounded planar harmonic mappings. We construct an explicit harmonic mapping $F_M$ satisfying the standard normalization conditions, which fails to be locally univalent strictly within the classical holomorphic Landau disk for all $M > M^* \\approx 4.451$. By combining this counterexample with the lower bound established by Ponnusamy, Kalaj, and Vuorinen (2014), we demonstrate that the exact asymptotics of the harmonic Landau radius is $\\fracπ{8M}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39676",
  "author_total": 1,
  "authors": [
    {
      "name": "Mikhail Borovikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ea7acd36m77pk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39676",
  "fulltext_url": null,
  "id": "ea7acd36m77pk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39676",
  "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": "W7215042245",
  "pdf_url": "https://arxiv.org/pdf/2609.39676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ea7acd36m77pk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius",
  "title_text": "A Counterexample to the Liu-Luo-Luo Conjecture on the Harmonic Landau Radius",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/ea7acd36m77pk",
  "venue": "arXiv",
  "versions": []
}