{
  "schema_version": 1,
  "problem_number": "AMR-022-6084",
  "title": "Root-Transform Bounds Do Not Force Logarithmic Coefficient Bounds",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct a bounded normalized univalent function whose logarithmic coefficients are not O(1/n), although every pth root transform satisfies the coefficient estimate O(n^(2/p-1)). This gives a negative answer to the converse question recorded as Hayman-Lingham Problem 6.84 (UnsolvedMath AMR-022-6084). The normalized logarithm of the same function is Holder continuous of every order below one on the closed disk; consequently every root transform has coefficients O(n^(-1+eta)) for every eta>0. Starting from the classical existence of arbitrarily large weighted logarithmic coefficients, the proof inserts dilated seeds by high-order root transforms and normalized composition. A congruence filter transfers selected coefficients while a summable diagonal budget preserves earlier coefficients and gives boundary regularity. This is an analytic existence construction using a credited classical seed theorem. The four-page English manuscript includes the complete proof and supporting exact rational checks. AI-assisted, self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CV",
    "math.CA"
  ],
  "keywords": [
    "AMR-022-6084",
    "univalent function",
    "logarithmic coefficients",
    "root transform",
    "Milin converse",
    "Holder regularity"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-6084/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-6084/paper.pdf?v=4855144ceb2b",
  "doi": "10.5281/zenodo.23655569",
  "zenodo_record_url": "https://zenodo.org/records/23655569",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "There exists one bounded normalized univalent function f with log(f/z) in C^alpha on the closed disk for every alpha<1, but sup_n n|gamma_n(f)| is infinite. For this same f, every root transform has c_n^(p)=O(n^(-1+eta)) for every eta>0 and therefore satisfies the source bound O(n^(2/p-1)) for all positive integers p. Complete negative answer to the converse in Hayman-Lingham Problem 6.84. One bounded normalized univalent function satisfies all root-transform bounds while its weighted logarithmic coefficients are unbounded. The classical large-coefficient seed theorem is credited; this is an analytic existence construction. 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": "4855144ceb2b9e8ba77d5920015fc6bc734be9ca78294baa2dabf3c933844b6b"
    },
    "source.zip": {
      "sha256": "fd7002ea6158c28fd77e3b535fbfc9c9334cbced7e4518e48c6b4fad96c8bce3"
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    "verification_report.md": {
      "sha256": "24c37ebbe6369119968995d7828521e67e99896bad3c0b519c50711c0ff9e03b"
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  },
  "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."
}