# Verification report — AMR-022-7023 (Hayman–Lingham, Problem 7.23, J. G. Krzyż: Kühnau's two-point condition and quasiconformal extension)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete negative answer** to the question as posed: Kühnau's two-point
condition (7.4), |u(z,ζ)| ≤ q(1−|z|²)⁻¹(1−|ζ|²)⁻¹, is sufficient for a quasiconformal extension **if and only
if q < 1/3**; in the range 1/3 ≤ q < 1 named in the problem it is not sufficient, for any q.
- **Counterexample (Theorem A).** g(z) = 1/(artanh z − i) is analytic, univalent and bounded in the disc,
  satisfies (7.4) with q = 1/3 (the supremum of (1−|z|²)(1−|ζ|²)|u| is exactly 1/3, not attained, approached only
  at the real diagonal up to automorphisms), and has no homeomorphic extension to the closed disc, hence no
  quasiconformal extension to the sphere or to the plane.
- **Positive side (Theorem B).** For q < 1/3, (7.4) gives univalence and a quasiconformal extension to the
  sphere with |μ| ≤ 3q (Ahlfors–Weill applied to the diagonal limit). No dilatation constant is claimed for an
  extension to C fixing ∞.
- **Sharpness (Proposition 1).** The best dilatation k*(q) that (7.4) guarantees satisfies
  q√(1 + 48q²/25) ≤ k*(q) ≤ 3q and k*(q) → 1 as q → 1/3⁻ (uses Kühnau's Grunsky inequality and the compactness of
  normalised K-quasiconformal maps, both cited, not re-proved).
- **Same-constant reading (Theorem C).** For every q ∈ (0,1) some f satisfies (7.4) with q and has no extension
  with |μ| ≤ q. **Not new**: this negative answer is due to Kühnau, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981)
  125–130 (Satz 2 and the remark on p. 126, which refers to Problem 7.23); Theorem C is a second proof with the
  family artanh(rz).
- **New, as far as the searches found:** only the computation q(strip map) = 1/3 and the resulting sharp
  threshold in the main reading. Prior art is not excluded (three unread candidates, below).

The note is unrefereed.

## Statement checked
- **Primary source.** W. K. Hayman, E. F. Lingham, *Research Problems in Function Theory*, Fiftieth Anniversary
  Edition, Springer 2019, doi:10.1007/978-3-030-25165-9; open version arXiv:1809.07200v2, Problem 7.23 (pp. 166–167
  of the arXiv version, PDF pages 167–168), proposed by J. G. Krzyż. Update 7.23 (2018): no progress reported. Read
  in a text copy of the arXiv version and, by the first and the third verification run, in the TeX e-print source.
  - 7.23 states that u (from Problem 6.57) is invariant under Möbius transformations and tends to {f, z}/6 on the
    diagonal; it recalls Kühnau's theorem (given there without citation): a quasiconformal extension to the whole
    plane with |f_z̄/f_z| ≤ q < 1 implies (7.4) (source label C7.1); it asks to show that (7.4) is also sufficient,
    "possibly with 1/3 ≤ q < 1".
  - Problem 6.57 prints u without the square on f(z) − f(ζ). This is so in the TeX source of arXiv:1809.07200v2
    itself (first verification run), so the misprint is in the source, not in the corpus extraction. Problem 6.58
    prints the square.
  - 6.57 is set in Σ(q) (exterior normalisation); its TeX also says "analytic and univalent in D" with
    f(z) = z + b₀ + b₁z⁻¹ + …, a normalisation slip of the source.
- **Corpus record.** AMR-022-7023 of ulamai/UnsolvedMath; statement = source text; listed as open
  (research classification OPEN-TRIAGE).
- **History.** Kühnau (1981, p. 126) refers to Problems 6.57, 6.58 and 7.23 of Anderson–Barth–Brannan, Bull.
  London Math. Soc. 9 (1977) 129–162; so the problem was in the 1977 list (that list was not read). The 2018 update
  reports no progress although Kühnau (1981) refers to the problem.

## Readings
- **(a) The square.** u(z,ζ) = f′(z)f′(ζ)/(f(z) − f(ζ))² − (z − ζ)⁻². Forced by the two properties stated in 7.23
  itself (Lemma 1 and Lemma 2(a); without the square the expansion begins −h⁻² + c₁h⁻¹ and has no limit, and
  Möbius invariance fails); with the square u is the classical kernel ∂²_{zζ} log[(f(z) − f(ζ))/(z − ζ)].
- **(b) Normalisation.** The disc normalisation printed in 7.23; the exterior (Σ) form of 6.57 is equivalent by
  the inversion identity u_{f(1/·)}(t,τ) = u_f(1/t,1/τ)/(t²τ²) and (|t|²−1)(|τ|²−1) = |t|²|τ|²(1−|1/t|²)(1−|1/τ|²)
  (Remark 1); G(t) = 1/artanh(1/t) = t − 1/(3t) + … is the counterexample in that normalisation.
- **(c) Whole plane.** A quasiconformal homeomorphism of the Riemann sphere agreeing with f on D (both (7.4) and
  extendability are invariant under Möbius post-composition; Kühnau 1974, p. 195, speaks of extensions to the
  whole sphere). The literal self-map-of-C reading fails trivially for unbounded f (e.g. z/(1 − z), u ≡ 0); this is
  not the answer given: the counterexample is bounded and has no homeomorphic extension to C or to the sphere, so
  the negative answer for 1/3 ≤ q < 1 holds in both readings; for q < 1/3 a bounded f also extends to C (Remark 4,
  no constant). In the C-reading the iff statement therefore holds for bounded f.
- **(d) Sufficient.** R1 (main): some quasiconformal extension; R2: an extension with |μ| ≤ q.
- **(e) The parenthesis.** Read as naming the range where sufficiency is asked (main reading): negative for
  every q in [1/3, 1) (Theorem A). Read as "possibly only for q < 1/3": the positive Theorem B. The iff statement
  (Corollary 1) settles both.
- **(f) Univalence** is not assumed; (7.4) forces it (Lemma 3).

## Results in the paper
| Item | Statement | Proof status |
|---|---|---|
| Lemma 1 | u_f(z,ζ) → S_f(z)/6 as ζ → z (with the square) | complete, elementary |
| Lemma 2 | invariance under Möbius post-composition, disc automorphisms (Q), dilation u_{f_r} = r²u_f(r·, r·) | complete |
| Lemma 3 | bounded Q_f forces injectivity | complete |
| Lemma 4 | sup over unit x of \|xᵀAx\| = ‖A‖ for bounded symmetric A | complete (polarisation, parallelogram law) |
| Section 2.1 | u_f = x(z)ᵀA x(ζ), ‖x(z)‖² = (1−\|z\|²)⁻²; Kühnau's inequality κ(f) ≤ k; q(f) ≤ κ(f) | Kühnau 1971 cited (as stated in Kühnau 1981, p. 126), not re-proved |
| Remark 1 | the Σ-normalised condition of 6.57 is equivalent | complete |
| Theorem A | g = 1/(artanh z − i): bounded, univalent, q(g) = 1/3 not attained, no homeomorphic extension | complete: closed form (3.1), partial fractions (3.2), cosine bound (3.3), elementary inequality H(b) ≤ 1/3 − 2b²/9 + 2b⁴/45 < 1/3 in steps (a)–(d), diagonal limit |
| Remark 3 | Q_g(z_j,ζ_j) → 1/3 forces, up to the automorphisms tanh(artanh z − t), convergence of both points to 0 | complete |
| Theorem B | q < 1/3 ⇒ univalent, qc extension to the sphere with \|μ\| ≤ 3q | Ahlfors–Weill (PAMS 13 (1962)) cited |
| Corollary 1 | (7.4) sufficient iff q < 1/3 | from A and B |
| Proposition 1 | q√(1+48q²/25) ≤ k*(q) ≤ 3q, k*(q) → 1 | Kühnau's inequality and Lehto–Virtanen, Ch. II §5 (compactness) cited, not re-proved |
| Theorem C | reading R2 negative for every q ∈ (0,1) | second proof of a result of Kühnau 1981; Kühnau's inequality cited |
| Remark 6 | at q = 1/3 every non-strip f maps onto a Jordan domain (Gehring–Pommerenke); whether these extend is left open | not claimed |

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **check_note.py** (standard library, < 1 s; output check_note.out, ALL CHECKS PASSED): exact rational checks of
  Lemma 1 (and the expansion without the square) on 25 random rational germs; Lemma 2(a),(c) and Remark 1 at
  rational points; S_artanh = 2/(1 − z²)²; the coefficients d_mn of artanh from its series (d₁₁ = 1/3,
  d₃₁ = d₁₃ = 4/45, d₂₂ = 1/30, d₅₁ = 44/945, d₃₃ = 83/2835, symmetry, parity); u_artanh(z,0) = 1/3 + 4z²/15 +
  44z⁴/189 + …; 81/2025 = 1/25 and r⁴/9 + 48r⁸/2025 = q²(1 + 48q²/25) at r² = 3q; the polynomial identities of
  steps (c) and (d). Floating point: the closed form (3.1) against the raw u at 300 pairs with |z₁ − z₂| ≥ 0.2,
  the sign of the partial fractions (3.2), H(b) and the bound cot b ≤ 1/b − b/3 on a grid, the bound of Remark 3,
  g(x) → 0 at ±1.
- **original/** (the programs with which the results were first obtained): symbolic_checks.py (sympy; all
  identities), numeric_checks.py (mpmath, 100 digits: Q_g from the raw definition of u at 19,995 random and
  adversarial pairs, max 0.333332845 < 1/3; the closed form agrees with the raw u to 1.2·10⁻⁵⁹; H(b) on 20,000
  points), grunsky_illustration.py (numpy FFT; 80 × 80 Grunsky sections, illustration only).
- **interval_certificate/** (from verification run B, unmodified except for the header comments): rigorous certificate that
  G(a,b) = cos²b |φ(a + ib)| < 1/3 for all (a,b) ≠ (0,0), |b| ≤ π/2: Region I (|a| ≥ 2) analytic bound
  1/4 + 1/sinh²2 = 0.32602; Region II (|Δ| ≤ 1) Taylor majorant on 17 polar boxes; Region III 3,506 boxes of
  adaptive interval subdivision; mpmath.iv, 30 digits, outward rounding. Detection tests reject false targets.
  Second, computer-assisted route to Theorem A(ii), independent of the inequality for H(b). Expected final line:
  `RESULT (threshold 0.3333333333, mode G): CERTIFIED G < threshold off the origin`.
- **verification_run_B/** (unmodified except for the header comments): b1_symbolic.py (14 symbolic checks), b3_numeric_attack.py (1,200
  Nelder–Mead searches from the raw u, maxima re-evaluated at 60 digits: 0.33333325; Koebe control gives 1),
  b4_grunsky.py (Grunsky coefficients by recursion, exact up to 12 × 12, double precision to 160 × 160).
- **Finite sections are lower bounds only.** For r = 0.5, 0.8, 0.9, 0.95, 0.99 the section norms of A_r are
  0.084, 0.237, 0.331, 0.407, 0.530 (80 × 80) and 0.0845, 0.2373, 0.3311, 0.4065, 0.5318 (160 × 160); at r = 1
  0.61 and 0.635. For artanh ‖Ae₁‖ ≥ 0.425 (160 × 160; 0.4215 at 80 × 80; the partial sums converge slowly — an
  earlier statement "tail like 1/log M" was wrong and was removed); the rigorous two-entry bound is √273/45 = 0.3672;
  the full value, from ‖Ae₁‖² = π⁻¹∬_{|Im W|<π/4}|φ(W)|² dσ_W (numerical quadrature), is 0.4352.
- **independent_run_2/** (third verification run, on the text of the note; see its README): r2_symbolic.py (43
  checks, ALL CHECKS PASSED), r2_rawQ.py (Q_g from the raw u at 20,000 adversarial pairs and in 160 hill climbs at
  precision raised with the proximity, up to 170 digits: all values < 1/3, smallest gap 2.5·10⁻⁵², raw vs closed
  form 1.8·10⁻⁴²), r2_Gcert.py (certificate of G < 1/3 by a different route: partial-fraction sum over
  0 < |k| ≤ 20 with a rigorous tail, own outward-rounded interval arithmetic, exact-rational majorant for
  |Δ| ≤ 1/2, analytic bound for a ≥ 4; soundness and detection tests), r2_grunsky.py (first column exactly and by
  series, the full column norm 0.4352, sections from a 2-D FFT of the raw u at 40 digits; all printed section
  norms confirmed).
- **Numerical caution.** Evaluating u from its definition in double precision near the diagonal is unreliable
  (a verification run saw spurious values up to 842 at a pair where 60-digit arithmetic gives 0.3333333331). At
  two points at distance h the raw formula loses about 3 log₁₀(1/h) digits at any precision (third run: at 60
  digits and h = 10⁻²⁰ the error is about 10⁻⁶). The raw-definition checks use 100 digits at distances ≳ 10⁻¹⁴,
  60 digits at distances ≥ 10⁻⁸, or (third run) a precision growing with 1/h; check_note.py uses the raw u only at
  well-separated pairs; the interval certificates use (3.1) or (3.2) with outward rounding.
- **Re-runs of 2026-10-20 from the package copies:** check_note.py, the three programs in original/, the three
  programs in interval_certificate/ (6 s, 33 s, < 1 s), b1_symbolic.py, b4_grunsky.py and
  `b3_numeric_attack.py g 1200 2` (37 s): every output byte-identical to the recorded one.
- **Re-runs by the third verification run** (from an extracted copy of source.zip): all eleven programs including
  `koebe 1200 2`: byte-identical outputs. After the edit of the header comments of six programs of run B, these
  were run again from the new archive: outputs unchanged.

## Independent verification runs
The results were first obtained with proofs and the programs in original/. Two independent verification runs,
both AI-assisted, followed (2026-10-20); each wrote its own programs. Run A examined the statement and all proofs;
run B made independent computations, re-ran the original programs, and checked the literature. The note was then
written, with Kühnau (1981) read in full, and a third independent run examined its text (section below).

| Item | Run A | Run B |
|---|---|---|
| Statement and readings against the source | CONFIRMED (TeX e-print read: the square is missing in 6.57 in the source itself) | CONFIRMED (misprint reading forced; both readings of the parenthesis to be stated) |
| Lemmas 1–4, Grunsky set-up | CONFIRMED (re-derived line by line) | CONFIRMED (re-derived; symbolic checks 14/14) |
| Theorem A | CONFIRMED (re-derived; 11,976 adversarial pairs at 60 digits, max 0.3333332; hill-climb on the closed form 0.33333332 at the real diagonal) | CONFIRMED (re-derived; rigorous interval certificate; optimisation attack, max 0.33333325) |
| Theorem B | CONFIRMED (sphere statement; C-version only a sketch) | CONFIRMED |
| Proposition 1, Theorem C | CONFIRMED modulo Kühnau 1971 and Lehto–Virtanen compactness (cited) | CONFIRMED given the same citations |
| Original programs | — | re-run from a copy: outputs byte-identical |
| Novelty | not established; prior art in Kühnau's 1974 notes possible | not established; new unread candidate Aksent'ev–Shabalin (2002) |

Neither run found a mathematical error.

**Corrections required by the runs and at the writing stage, all applied in the note:**
1. (A) Zbl 0315.30029: "vol. 24 (1974), no. 6, published 1975".
2. (A) Explicit statement that the TeX source of 6.57 lacks the square, that 7.23's own assertions force it, that
   6.58 prints it, and that it is the classical kernel.
3. (A) Theorem B everywhere as "a quasiconformal extension to the sphere with |μ| ≤ 3q"; no constant for C.
4. (A) Pommerenke 1975: the book could not be consulted; no theorem numbers (page attribution corrected by the
   third run, item 17 below).
5. (A) Inversion remark (Remark 1).
6. (A) Novelty wording: only q(strip map) = 1/3 and the sharp threshold; prior art not excluded.
7. (B) Aksent'ev–Shabalin (2002) added to the unread candidates.
8. (B) Finite-section numbers worded as numerical lower bounds (0.425 at 160 × 160; rigorous √273/45).
9. (B) One sentence on both readings of the parenthesis.
10. (B) Numerical caution on double-precision evaluation of u.
11. (Writing stage, after reading Kühnau 1981 in full) Kühnau 1981 credited prominently (Satz 2 and p. 126 answer reading R2; Theorem C is a second proof;
    reading R1 is not treated there); Kühnau, Ann. UMCS 22–24 (1968–70), Zhuravlev 1978 and Becker–Pommerenke 1978
    cited as reported there, next to Theorem B; the 2018 update noted neutrally; dating by the 1977 list; Kühnau
    1974 p. 195 cited only for what it shows; the unread list fixed.

### Third verification run (on the text of the note)
A third independent verification run, AI-assisted, examined the text of the note after it was written, as if
nothing had been checked. It read Problems 6.57, 6.58 and 7.23 again in the TeX source of arXiv:1809.07200v2 and
in the PDF, read Kühnau (1981) again in full and p. 195 of Kühnau (1974) from page images, checked all twelve
printed DOIs in Crossref and the zbMATH records 0226.30021, 0283.30015, 0315.30029, 0454.30016 (and 0248.30021),
re-derived every proof and lemma line by line, wrote four programs of its own (independent_run_2/), re-ran the whole
package from the archive, and repeated part of the literature search (arXiv API, zbMATH documents citing Kühnau
1981 and 1974, Crossref, OpenAlex — citing lists refused with HTTP 429 —, one web search).

| Item | Third run |
|---|---|
| Statement, readings, scope | CONFIRMED; page numbers of the source and two sentences of the abstract corrected |
| Lemmas 1–4, §2.1, Remark 1 | CONFIRMED (re-derived; 43 symbolic or 60-digit checks) |
| Theorem A, Remarks 2, 3 | CONFIRMED (re-derived; raw Q_g at adaptive precision, all < 1/3; second certificate by a different route) |
| Theorem B, Corollary 1, Remark 4 | CONFIRMED |
| Proposition 1, Theorem C | CONFIRMED given Kühnau 1971 and Lehto–Virtanen (cited) |
| Attribution to Kühnau 1981 | CONFIRMED (Satz 2 works for every q ∈ (0,1): any r with r⁴ ≤ q < r²) |
| Computations | all printed numbers confirmed; the statement "tail like 1/log M" was wrong (removed) |
| Package | all outputs reproduced byte for byte |
| Novelty | no printed answer to reading R1 found; later work on the Grunsky condition named as related |

It found no mathematical error. **Corrections required by the third run, all applied:**
12. Problem 7.23 is on pp. 166–167 of the arXiv version (printed page numbers), not pp. 167–168.
13. Abstract and §1.2(c): for extensions to the plane the answer is the same for bounded f only (unbounded f have
    none); the negative answer for 1/3 ≤ q < 1 holds in both readings.
14. Abstract: no printed answer was found in the main reading R1 (R2 was answered by Kühnau 1981).
15. §6: "its tail decays slowly, like 1/log M" removed (numerically T(M)·log M decreases and T(M)·log³M increases,
    T(M) = ‖Ae₁‖² − ‖Ae₁‖²_M); the identity ‖Ae₁‖² = π⁻¹∬_{|Im W|<π/4}|φ(W)|² dσ_W and its numerical value 0.4352 added.
16. Bibliography: Kühnau, Bull. Soc. Sci. Lett. Łódź, dated 1975 (vol. 24 (1974), no. 6); Kühnau, Ann. UMCS 22–24:
    published 1972, Zbl 0248.30021.
17. §2.1: Kühnau cites Pommerenke's book for the inequality without pages; pp. 290–291 are cited for the argument of
    his proof of Satz 1 and p. 292 for the sufficiency question.
18. Related work: Kühnau, Comment. Math. Helv. 61 (1986) 290–307 (doi:10.1007/BF02621917, Zbl 0605.30023) and
    Krushkal, Sib. Math. J. 28 (1987) 104–110 (doi:10.1007/BF00970217, Zbl 0624.30020): when the full Grunsky
    condition with κ = q suffices for a Q-qc extension; known from their zbMATH reviews; not about (7.4).
19. Verification paragraph, §7 and this report: the third run recorded; the digit-loss rule in the numerical
    caution; independent_run_2/ added.
20. Header comments of six programs of run B: internal role words replaced by "Verification run B" and "the
    original programs"; outputs unchanged.

## Relation to the literature, novelty and scope
- **Read.** Problems 6.57, 6.58, 7.23 and updates of Hayman–Lingham (arXiv v2; TeX source by run A). Kühnau,
  "Zu den Grunskyschen Coeffizientenbedingungen" (title as printed on p. 125; running head and Crossref:
  "Koeffizientenbedingungen"), Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981) 125–130, doi:10.5186/aasfm.1981.0608,
  Zbl 0454.30016, in full (open access; read from page images). Content used: conditions (4)–(7) for Σ₀ (all follow
  from (4) with the same constant; (7) bounds the kernel U at finitely many points); Satz 1 (κ = 1/λ, Fredholm
  eigenvalue, best possible); Satz 2 (maps satisfying (4)–(7) with κ = q but without Q-qc extension: Teichmüller
  maps from an ellipse via the chain z^{3/2}, 𝔷 + r²𝔷̄, 𝔴^{2/3}, extremal dilatation q* = r², 1/λ < q*); p. 126: the
  same for other consequences of (4), e.g. (6) of Kühnau 1974, with reference to 6.57, 6.58, 7.23 of the 1977
  list; Pommerenke's converse (κ < 1), Kühnau UMCS ((1+3κ)/(1−3κ) for κ < 1/3), Zhuravlev (qualitative analogues),
  Becker–Pommerenke. Kühnau, Math. Nachr. 59 (1974) 195–198, only p. 195: Kraus's bound 6/(1−|z|²)², 6q for maps
  with Q-qc extension to the whole sphere, the converses with 2 (Nehari) and 2q, further contributions announced.
- **Cited as reported and not read:** Kühnau 1968–70 (UMCS), Zhuravlev 1978, Anderson–Barth–Brannan 1977,
  Becker–Pommerenke 1978; Kühnau 1971 and Pommerenke 1975 for the statements used. **Cited as standard, not
  re-read:** Ahlfors–Weill 1962, Nehari 1949, Gehring–Pommerenke 1984, Lehto–Virtanen 1973.
- **Searches (2026-10-20).** The 2018 update; zbMATH Open API (all Kühnau records, Krzyż's records, keyword
  searches, documents citing the two 1974 notes and, third run, those citing Kühnau 1981); arXiv API; Crossref;
  OpenAlex and Semantic Scholar (partly refused, HTTP 429); a few web searches.
- **Unread prior-art candidates:** Kühnau, Math. Nachr. 59 (1974) 195–198 (Zbl 0283.30015,
  doi:10.1002/mana.19740590113; only p. 195 read; its inequality (6) is very likely (7.4), unchecked); Kühnau,
  Bull. Soc. Sci. Lett. Łódź 24 (1974) no. 6, published 1975, dated 1975 in zbMATH (Zbl 0315.30029); Aksent'ev–Shabalin, Handbook of
  Complex Analysis: Geometric Function Theory, Vol. 1 (2002) 169–206, doi:10.1016/S1874-5709(02)80009-9
  (Zbl 1067.30034; cites Kühnau 1974; full text not accessible).
- **Related, different:** Shen Yuliang, Acta Math. Sin. 17 (2001) 397–402 (the same kernel, integral norm
  only); two-point distortion theorems of Chuaqui–Duren–Ma–Mejía–Minda–Osgood (Pacific J. Math. 254 (2011)) and
  Ma–Mejía–Minda (a different quantity); later work on the same-constant question for the full Grunsky condition:
  Kühnau, Comment. Math. Helv. 61 (1986) 290–307, Krushkal, Sib. Math. J. 28 (1987) 104–110 (zbMATH reviews only;
  documents citing Kühnau 1981 in zbMATH, third run).
- **Bibliographic data** verified with Crossref on 2026-10-20: Hayman–Lingham, Anderson–Barth–Brannan,
  Ahlfors–Weill, Nehari, Gehring–Pommerenke, Kühnau 1971, 1974 (Math. Nachr.), 1981, Lehto–Virtanen,
  Becker–Pommerenke, Aksent'ev–Shabalin, Chuaqui et al.; Crossref lists two DOIs for Shen (2001), so none is given.
- **Conclusion.** No printed answer to the question in reading R1 was found in the sources searched; a search
  that finds nothing is not a proof of novelty. No priority is claimed. Reading R2 was answered negatively by
  Kühnau (1981).

## 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/