{
  "schema_version": 1,
  "problem_number": "AMR-022-5039",
  "title": "Duren's Problem on Integral Means of the Derivative of a Subordinate Function: the Radius 1/2 for p ≤ 2 and Bounds for p > 2",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let g be subordinate to f in the unit disc, that is, g = f∘φ with φ analytic, |φ| < 1 and φ(0) = 0. Goluzin proved in 1951 that M_2(r,g′) ≤ M_2(r,f′) for r ≤ 1/2, and that M_p(r,g′) ≤ M_p(r,f′) for every p > 0 when r ≤ √2 − 1. Problem 5.39 of Hayman and Lingham's Research Problems in Function Theory, posed by P. L. Duren, asks for the largest number r_p such that the inequality between the p-th means of the derivatives holds for 0 < r < r_p. We show that r_p = 1/2 for every 0 < p ≤ 2, in the quantitative form M_p(r,g′) ≤ (α² + 4r²(1 − α²))^{1/2} M_p(r,f′) for r ≤ 1/2, where α = |φ′(0)|. The proof is short: Hölder's inequality between Littlewood's subordination theorem and Goluzin's theorem, applied to a power of the zero-free part of f′. For p > 2 the problem remains open, and we prove two-sided bounds. The function p ↦ r_p is non-increasing and left-continuous, r_p → 1/2 as p ↓ 2, and r_p → √2 − 1 = r_∞ as p → ∞. Moreover r_p ≥ r_1(p) > √2 − 1 for every finite p > 2, where r_1(p) is the root in (√2 − 1, 1/2) of 8r⁴ − 16r³ + (p+4)r² + 2pr − p = 0; so the lower bound √2 − 1 recorded with the problem is not best possible for any finite p. In the other direction, r_p < 1/2 for every p ≥ 12.0068, with explicit upper bounds for larger p, for instance r_20 < 0.4779 and r_100 < 0.4372; these rest on 19 explicit pairs (f, φ) violating the inequality, verified in exact rational arithmetic by three independently written programs. The exact value of r_p for p > 2 is not determined; numerical experiments, which prove nothing, suggest that r_p = 1/2 up to p ≈ 12.0065. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CV",
    "math.CA"
  ],
  "keywords": [
    "subordination",
    "integral means",
    "derivative of a subordinate function",
    "Goluzin's theorem",
    "Littlewood's subordination theorem",
    "Rogosinski's coefficient inequality",
    "Schwarz function",
    "Schwarz–Pick lemma",
    "exact rational arithmetic",
    "partial answer",
    "Research Problems in Function Theory",
    "Problem 5.39",
    "Duren",
    "UnsolvedMath",
    "AMR-022-5039",
    "math.CV",
    "math.CA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-09",
  "publication_date": "2026-10-09",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-09",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-5039/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-5039/paper.pdf?v=d11e7c882da6",
  "doi": "10.5281/zenodo.23252438",
  "zenodo_record_url": "https://zenodo.org/records/23252438",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A partial answer to Problem 5.39 (P. L. Duren) of Hayman and Lingham's Research Problems in Function Theory: the largest radius r_p for which the p-th integral means of the derivative do not increase under subordination. Proved: r_p = 1/2 for every 0 < p ≤ 2; r_p is non-increasing and left-continuous with limits 1/2 (p ↓ 2) and √2 − 1 (p → ∞); an explicit lower bound r_1(p) > √2 − 1 for p > 2; and r_p < 1/2 for every p ≥ 12.0068, with explicit upper bounds for larger p, from 19 certificates in exact rational arithmetic checked by three independently written programs. The exact value of r_p for p > 2 stays open; the statement that r_p = 1/2 up to p ≈ 12.0065 is numerical evidence only. The result for p ≤ 2 has a short proof built from tools available in 1951 (Littlewood, Goluzin, Hölder) and may be known to experts: Goluzin's 1951 papers, read in full, do not contain it, but Goluzin's book, the books of Duren, Pommerenke and Pavlović, and MathSciNet were not checked. Three independent AI-assisted verification runs checked the proofs. Unrefereed; no priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "d11e7c882da6f2557aa7b8a77dab7590500b0a32fb193721f70d040cfc0cc1a3"
    },
    "source.zip": {
      "sha256": "fe8f9f4f1c2a231c1762f6234a1f8147182629dd7a23f118a8900ceab9187c1f"
    },
    "verification_report.md": {
      "sha256": "8e9dadc63eef0deb11d5d91e2d62e9ab294cf591d9fe9e9c65b2ab27ae5a0da5"
    }
  },
  "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."
}