{
  "schema_version": 1,
  "problem_number": "AMR-022-6050",
  "title": "A Counterexample to the Area Inequality for Geometric Means of Exterior Univalent Functions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A question of D. K. Thomas, recorded as Problem 6.50 in Hayman and Lingham's Research Problems in Function Theory, asks whether geometric means of normalized exterior univalent functions satisfy the area-theorem coefficient inequality. We give an explicit counterexample at exponent 1/2. The two factors are odd, zero-free and univalent on |z|>1, and their normalized geometric mean is analytic throughout that domain. Its first four nonzero negative Laurent coefficients have weighted squared sum 269463/262144>1. A fixed radial dilation by 1001/1000 gives factors and a mean analytic across the unit circle, with full coefficient sum finite and greater than 101/100. The proof uses elementary conformal maps and exact rational arithmetic. This gives a negative answer to the entire single-question frozen UnsolvedMath record AMR-022-6050. The package includes the complete four-page proof, exact and symbolic checks, and a source/scope audit. AI-assisted self-audited preprint; no independent peer review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CV"
  ],
  "keywords": [
    "AMR-022-6050",
    "Thomas problem",
    "geometric mean",
    "exterior univalent functions",
    "area theorem",
    "Koebe function"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-6050/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-6050/paper.pdf?v=d6d43acd417b",
  "doi": "10.5281/zenodo.23649631",
  "zenodo_record_url": "https://zenodo.org/records/23649631",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "An explicit pair of odd zero-free exterior univalent functions f(z)=z-z^-1 and g(z)=z B(z^-2), B(w)=((1-i)sqrt(1-w^2)+(1+i)(1-iw))/2, has an analytic normalized geometric mean with exponent1/2 whose first four nonzero Laurent coefficients give weighted squared sum269463/262144>1. Radial dilation by1001/1000 makes both factors and the mean analytic across the unit circle, with full weighted sum finite and greater than101/100. This gives a negative answer to Hayman-Lingham Problem 6.50, including zero-free odd factors and a mean analytic across the unit circle. Classical conformal mapping tools are credited; no absolute-priority claim is made. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "d6d43acd417b7d09e00042e266f33b55f9fced108dfb3ea81e91bae11932bade"
    },
    "source.zip": {
      "sha256": "5217ccb1d0bc1e77a3776dc677454d62420a34e2c43b4538dc4d2ee8174d70ac"
    },
    "verification_report.md": {
      "sha256": "bb0cd97a51000687da95bea68f2897b16a38fa96cb9d1abf6082166a526a83b2"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}