# Verification report — AMR-022-5039 (Hayman–Lingham Problem 5.39, posed by P. L. Duren)

Verification date: 2026-10-09.

**Verdict.** Partial answer. Let g ≺ f in the unit disc (g = f∘φ, φ analytic, |φ| < 1, φ(0) = 0) and let r_p be the
largest number such that M_p(r,g') ≤ M_p(r,f') for 0 < r < r_p. The note proves r_p = 1/2 for every 0 < p ≤ 2, with
the quantitative form M_p(r,g') ≤ (α² + 4r²(1 − α²))^{1/2} M_p(r,f') for r ≤ 1/2, α = |φ'(0)|. For p > 2 it proves
two-sided bounds: r_1(p) ≤ r_p ≤ 1/2 with r_1(p) > √2 − 1 the root of 8r⁴ − 16r³ + (p+4)r² + 2pr − p = 0 in
(√2 − 1, 1/2); r_p < 1/2 for every p ≥ 12.0068; explicit upper bounds for larger p. **The exact value of r_p for
p > 2 remains open.** Theorems A–C do not depend on any computation; Theorem D rests on 19 strict inequalities
certified in exact rational arithmetic by three independently written programs. The note is unrefereed.

## Statement checked
- **Primary source.** W. K. Hayman and E. F. Lingham, *Research Problems in Function Theory*, Fiftieth Anniversary
  Edition, Springer 2019 (arXiv:1809.07200), Problem 5.39, printed pages 99–100 of the arXiv version (the definition
  of subordination is on page 98). The arXiv version was read.
  - The problem recalls Goluzin's inequality M_2(r,g') ≤ M_2(r,f') for 0 ≤ r ≤ 1/2 and the example f = z, g = z²,
    and asks for the largest r_p (0 < r_p < 1, independent of f and g) such that M_p(r,g') ≤ M_p(r,f') for
    0 < r < r_p whenever g ≺ f. It is attributed to P. L. Duren.
  - Subordination is defined just before Problem 5.38: g = f∘φ with φ analytic in the disc, φ(D) ⊂ D, φ(0) = 0.
    No univalence and no normalisation is imposed in Problem 5.39; p ranges over all positive numbers.
  - A note states |g'(z)| ≤ |f'(φ(z))| for |z| ≤ √2 − 1 (with a reference to Carathéodory), derives r_p ≥ √2 − 1 for
    all p from Littlewood's theorem, and says that this value need not be optimal. The update reports no progress.
- **Corpus record.** ulamai/UnsolvedMath (version 1.6.0), record AMR-022-5039, labelled open. Its statement agrees
  with the source; it only replaces an equation number of the source by a label.
- **Formal reading used.** 𝓡_p = set of r ∈ (0,1) such that the inequality holds for all analytic f and all Schwarz
  functions φ; r_p = sup{ρ : (0,ρ) ⊂ 𝓡_p}. The paper shows 𝓡_p = (0, r_p] (Lemma 2.5), so open or closed ranges of r
  make no difference.

## Readings
| Reading | Result |
|---|---|
| All analytic f, all Schwarz functions φ, 0 < p < ∞ (the problem as posed) | r_p = 1/2 for 0 < p ≤ 2; r_1(p) ≤ r_p ≤ 1/2 for p > 2; r_p < 1/2 for p ≥ 12.0068; exact value open for p > 2 |
| p = ∞ (not asked; included for comparison) | r_∞ = √2 − 1 (classical in substance) |
| f restricted to the class S of normalised univalent functions | Lower bounds unchanged; r_p = 1/2 for p ≤ 2 unchanged (f = z ∈ S); all 19 certificate functions f lie in S, so the violations hold within S at the exponents of Table 2. The extension to larger exponents (via monotonicity in p) is proved only for the unrestricted problem. |
| φ required to be univalent | A different problem (the maps z² and φ_a are excluded); not treated |
| Pointwise majorisation |g'| ≤ |f'| (Problem 5.38) or area means (Goluzin's Theorem 4, Reich) | Different problems; not treated |

## Results in the paper
- **Lemmas 2.1–2.3 and 2.5** (reductions): zero-free majorant; passage to the disc algebra; Hilbert space form (the inequality
  for all f is equivalent to ∫|K∘φ|²|φ'|^p ≤ ∫|K|² on |z| = r for all K in the disc algebra of |z| < r); the set of
  admissible radii is (0, r_p] with √2 − 1 ≤ r_p ≤ 1/2.
- **Lemmas 3.1–3.4** (tools, each with a complete proof): Littlewood's theorem for the exponent 2; a refined
  Rogosinski inequality Σ_{n≤N}|b_n|² ≤ Σ_{n<N}|a_n|² + |φ'(0)|²|a_N|²; a quantitative form of Goluzin's theorem
  with the factor γ = α² + 4r²(1 − α²) for r ≤ 1/2; two consequences of the Schwarz–Pick lemma. No novelty is
  claimed for these lemmas.
- **Theorem A.** For 0 < p ≤ 2 and r ≤ 1/2: M_p(r,g') ≤ γ^{1/2} M_p(r,f') ≤ M_p(r,f'); the radius 1/2 is sharp
  (f = z, g = z²). Hence r_p = 1/2. Proof: Hölder's inequality between Lemma 3.3 and Lemma 3.1, applied to a power
  of the zero-free part of f'.
- **Theorem B.** p ↦ r_p is non-increasing and left-continuous; r_∞ = √2 − 1 = lim_{p→∞} r_p; lim_{p↓2} r_p = 1/2.
  Consequently r_p = 1/2 exactly for p ≤ p_*, with 2 ≤ p_* < 12.0068 (using Theorem D).
- **Theorem C.** For p > 2: r_p ≥ r_1(p) > √2 − 1; r_1(p) = 1/2 − (p−2)/8 + O((p−2)²) and
  r_1(p) = √2 − 1 + (92 − 65√2)/p + O(p⁻²). So the bound √2 − 1 of the source is not best possible for any finite p.
- **Theorem D.** r_p < 1/2 for p ≥ 12.0068; r_p < r_0 for p ≥ p_0 for fifteen further pairs (p_0, r_0), for
  instance (14, 0.4932), (20, 0.4779), (100, 0.4372), (1000, 0.4207); for f = z alone the inequality fails at
  r = 1/2 when p = 16.
- **Not proved (Section 8 of the paper).** Numerical evidence and Conjecture 8.1: r_p = 1/2 up to p_1 = 12.0065…,
  the threshold of the family φ_a(z) = z(z+a)/(1+az), and r_p equal to the critical radius of that family for
  larger p. The paper states the ranges that were searched and that these searches prove nothing.

## Computations (scripts and outputs in reproducibility/)
- **What is certified.** For each of the 19 rows of Table 2 (rational data p, r, a, h), a strict inequality between
  two explicitly computed rational numbers which, by Lemma 7.1 (even p) or Lemma 7.2 (rational p), implies
  M_p(r,g') > M_p(r,f') for g = f∘φ_a, f' = h.
- **Program 1** (`certificates/c1_cert_exact.py`, the first exact program of the note; standard library only; about
  2 s). Integers and `Fraction` only in all comparisons. Output: 19 certified violations. A negative control
  (`certificates/negative_control.py`) returns no certificate on four non-violating data sets and a certificate on
  one violating data set.
- **Program 2** (`independent_run_2/v2_cert.py`, written independently in the second verification run). Different
  algorithm: integer recurrence from a first-order differential equation, truncation orders 120–4000; it also
  compares the Taylor coefficients T_n, n ≤ 120, obtained by two exact methods. Output: all 19 violations
  certified; its lower bounds agree with those of Program 1 in the eight decimals printed in Table 2.
- **Program 3** (`independent_run_3/r3_exact.py`, written in the third verification run before Programs 1 and 2
  were read; standard library only; about 4 s). It reads the data from the LaTeX table. Even p: composition of power
  series, termwise differentiation, k-th power with integers. Rational p: T = exp((p/2) log S), with log S assembled
  from the logarithms of the linear factors and from log h composed with φ_a. Output: all 19 violations certified,
  with the eight decimals of Table 2 in every row, also for the truncation orders 640 / 200 / 480; five control data
  sets are not certified.
- **Cross-checks (floating point).** Double-precision quadrature of the 19 ratios (`n5_crosscheck.py`, agreement to
  ten digits) and two quadratures with 50 digits (`v2_quad.py`, `r3_quad.py`): every ratio exceeds 1 and exceeds the
  printed bound by less than 10⁻⁸.
- **Table of r_1(p).** `certificates/r1_table.py` verifies the rounding of every printed value in exact rational
  arithmetic; the first verification run verified it by exact root isolation, the third by exact sign tests of the
  quartic for all 24 values printed in the paper.
- **Consistency checks of the proved statements** (floating point or symbolic; not proofs): `numerics/n1_sanity.py`,
  `n7_direct_check.py`, `s1_symbolic_checks.py`, `independent_run_3/r3_checks.py`.
- **Re-runs on 2026-10-09.** The three exact programs, the quadratures, all scripts of `numerics/` (except the
  exploratory ones) and several scripts of the verification runs were run again, the last time from an extracted
  copy of the source archive; the outputs are identical to the recorded ones apart from timing lines (list in
  `reproducibility/README.md`).

## Independent verification runs
Three independent verification runs, all AI-assisted. The first two examined the working manuscript from which the
paper was written (2026-10-08/09); the third examined the finished paper (2026-10-09).

| Item | Run | Verdict |
|---|---|---|
| Statement of the problem against the source and the corpus record | 1, 2, 3 | CONFIRMED (run 3: one paraphrase made exact) |
| Lemmas 2.1–2.3, 2.5, 3.1–3.4 | 1, 3 | CONFIRMED (every proof re-derived) |
| Theorem A (and its quantitative form), Remark 4.1 | 1, 3 | CONFIRMED (run 3: the reason in Remark 4.1 written out) |
| Theorem B | 1, 3 | CONFIRMED |
| Theorem C | 1, 3 | CONFIRMED (run 1: theorem and proof correct; a remark after it was inaccurate and was rewritten; run 3 confirmed the rewritten Remark 6.2) |
| Lemmas 7.1, 7.2 and Theorem D | 2, 3 | CONFIRMED (two further exact programs; no fix needed) |
| Tables 1, 2, 3 | 2, 3 | CONFIRMED (run 3: every entry recomputed) |
| Numerical evidence and conjecture (labelling) | 2, 3 | Correct in substance; run 2: six inaccuracies of wording; run 3: two more sentences |
| Literature | 2, 3 | Both parts of Goluzin's 1951 paper read in full; Dieudonné's pages 351–353 compared; no prior statement of Theorems A–D found |

Run 1 also checked Lemma 3.2 and the finite form of the chain of Lemma 3.3 in exact rational arithmetic (541,680
comparisons, no failure) and searched for counterexamples to Theorems A and C (none found; three control cases with
known violations were detected). Run 2 recomputed the numerical evidence with its own code. Run 3 read the final
text completely, re-derived every proof, wrote the third exact program and its own code for the tables and for the
numbers of Section 8, ran the package again from its source archive, compared the cited statements with the sources
and verified the DOIs of the references through Crossref. No run found a mathematical error.

All required fixes of runs 1 and 2 were applied:
1. The remark after Theorem C (now Remark 6.2) was rewritten: the argument loses in two places, the second being
   that the two Schwarz–Pick bounds are sharp in opposite configurations. An improved threshold suggested by
   numerical experiments is mentioned only as an unproved observation.
2. Lemma 2.1 (zero-free majorant): the proof of continuity up to the boundary was rewritten (limit 0 at boundary
   zeros; local branch of the logarithm elsewhere).
3. Lemma 2.5 (admissible radii): a note on the forward reference to Lemmas 3.1 and 3.4(a) was added.
4. Lemma 2.3 (Hilbert space form): the application of Lemma 2.1 to a polynomial K, possibly with zeros on the
   circle, is now explicit.
5. Goluzin's 1951 paper: the paper now states what it contains (Theorem 5: p = 2, r ≤ 1/2; Theorem 6: all p > 0,
   r ≤ √2 − 1, with the remark on 1/2 < r < 1), attributes the note of Problem 5.39 to Goluzin's Theorem 6, and
   warns that the zbMATH display of the review of this paper omits the exponent 2.
6. Section 8: the range of the parameter a in the maximum is given (0.3 ≤ a ≤ 0.95; the end points a = 0 and a → 1
   give the trivial value 1); the exploratory runs with degree 4 are described correctly (p ∈ {11, 11.9}); "local
   optimality" was replaced by "first-order test in seven directions"; the conjecture for p > p_1 is said to rest on
   the family computation, with the additional tests of run 2 listed; the remark about search queries without hits
   was removed.
7. Script docstrings now refer to the numbering of the paper; an unused helper was removed from the first exact
   program (the executed code and the output are unchanged).

The optional quantitative form of Theorem A (suggested by run 1) was added.

All required fixes of run 3 were applied (none changes a theorem, a proof, a table entry or a number):
1. Programs, computations and searches are no longer attributed to a person; the first exact program is named by
   its file name.
2. The third run and its exact program are recorded (Sections 7 and 10, Verification paragraph, abstract).
3. Section 1.1 gives the note attached to the problem as it stands in the source (|g'(z)| ≤ |f'(φ(z))| for
   |z| ≤ √2 − 1), then its meaning |φ'| ≤ 1.
4. Remark 4.1: the reason why equality forces f' ≡ 0 when φ is not a rotation is written out; the case of constant
   f is named.
5. Section 8.4: the sentence that inferred a slow decay of r_p − (√2 − 1) from the critical radii of the family was
   replaced. Those radii bound r_p from above; the order of decay of r_p − (√2 − 1) is not known.
6. Section 8.3: the searches are said to concern exponents p ≤ p_1, and the outcome of the repeated first-order
   test is stated (the maximum decreased in every case).
7. Part II of Goluzin's paper is cited and described (pointwise majorisation of univalent subordinate functions;
   nothing on integral means).
8. Scope paragraph: dates and number of the searches; "Goluzin's paper has the squares".
9. This report, the README and the deposit metadata were brought to the final state.
Small precisions in the proof of Lemma 3.2 (the case N = 2; Goluzin's use of the case N = 1), in Remark 6.2 (the
point where |φ'| is maximal) and in the definition of r_c(p) were made as well.

## Relation to the literature, novelty and scope
- **Read for this note.** Problem 5.39 and its update in the arXiv version of Hayman–Lingham; Goluzin's paper
  "On majorants of subordinate analytic functions. I", Mat. Sbornik N.S. 29(71) (1951), 209–224, completely, in the
  scan of the Göttingen Digitisation Centre (pages 214–223 at reduced resolution), and its second part, ibid.,
  593–602, completely, in the same scan; the introduction and the main theorem of E. Reich, Pacific J. Math. 4
  (1954), 259–274; pages 351–353 of J. Dieudonné's memoir, Ann. Sci. École Norm. Sup. (3) 48 (1931), 247–358 (the
  estimate of |φ'| of Lemma 3.4(a), the bound |φ'| ≤ 1 for |z| ≤ √2 − 1, the sharp bound beyond and its extremal
  function); the zbMATH reviews of Rogosinski (1943), Reich (1954), Shah (1957) and of the first edition of
  Goluzin's book; the zbMATH summary of Dmitrović–Karapetrović (2024).
- **What Goluzin's 1951 paper contains.** Part I. Theorem 1: Littlewood's theorem (new proof, by dividing by the
  Blaschke product of the disc |z| < r). Theorem 2: Rogosinski's inequality. Theorem 3: the weighted form.
  Theorem 4: areas, r ≤ 1/√2. Theorem 5: p = 2, r ≤ 1/2, equality only for rotations when r < 1/2. Theorem 6: all
  p > 0, r ≤ √2 − 1; remark: fails for 1/2 < r < 1 (pair z, z²). Theorems 7–10: pointwise majorisation. No statement
  for p ≠ 2 at radii above √2 − 1. The note of Problem 5.39 is Goluzin's Theorem 6. Part II: four theorems on
  pointwise majorisation of univalent subordinate functions (|f| ≤ |F| in |z| < 0.39…, |f'| ≤ |F'| in
  |z| < 3 − 2√2, p-fold symmetric and meromorphic versions), by Löwner's method; nothing on integral means.
- **Caution.** The review of part I displayed by zbMATH (Zbl 0044.30501), a machine conversion of a scanned
  review, shows the integrals of Theorem 5 without the exponent 2. Goluzin's paper has the squares; the display is
  not a statement for p = 1.
- **Searches (8 and 9 October 2026, UTC).** arXiv API, zbMATH Open, Crossref, OpenAlex and eight web searches (for
  the note and in its three verification runs). They found no statement of Theorem A or of Theorems B–D. Sites
  that refused automated access or returned nothing (mathnet.ru, the publisher's page of Mathematika, two AMS
  pages for Goluzin's book, the De Gruyter page of Pavlović's book, Encyclopedia of Mathematics) were not worked
  around; a title-and-author search of the Internet Archive for Duren's *Univalent Functions* returned no item.
- **Not checked.** Goluzin's book *Geometric Theory of Functions of a Complex Variable* (Chapter VIII); Duren's books
  *Univalent Functions* and *Theory of H^p Spaces*; Pommerenke's *Univalent Functions*; Pavlović's books; the papers
  of Littlewood (1925), Rogosinski (1943), Shah (1957) and Carathéodory's book (the forms of the classical results
  that are used are proved in the paper); the rest of Dieudonné's memoir; MathSciNet.
- **Novelty.** Theorem A has a short proof (Hölder's inequality between Littlewood's theorem and Goluzin's theorem,
  applied to a power of the zero-free part of f'), built from tools available in 1951. An earlier statement, in
  particular of Theorem A, cannot be excluded; no priority is claimed, and Theorem A may be known to experts.
  Lemmas 3.2 and 3.3 are stated with proofs and without a claim of novelty. This negative search is not a proof of
  priority.
- **Scope.** r_p is determined for 0 < p ≤ 2 and bounded from both sides for p > 2. Open: the exact value of r_p
  for p > 2; the number p_* ∈ [2, 12.0068); Conjecture 8.1; right-continuity and strict monotonicity of p ↦ r_p.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/