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Brennan's conjecture

Let W⊂CW\subset\mathbb C be a simply connected domain whose boundary in the Riemann sphere has at least two points, and φ:W→D\varphi:W\to\mathbb D a conformal bijection onto the unit disk. Brennan (1978) conjectured that ∫W∣φ′∣s dA<∞\int_W|\varphi'|^s\,dA<\infty for every 4/3<s<44/3<s<4; the Koebe function shows the range cannot be enlarged. Equivalently, for the inverse map ff, ∫D∣f′∣t dA<∞\int_{\mathbb D}|f'|^{t}\,dA<\infty for −2<t<2/3-2<t<2/3, and in integral-means language BS(−2)=1B_{\mathcal S}(-2)=1. Brennan proved the range up to a bit beyond s=3s=3; Carleson-Makarov, Pommerenke, Bertilsson (s<3.421s<3.421), Hedenmalm-Shimorin and Sola (B(−2)≤1.206B(-2)\le1.206) improved the bounds. Does the full range 4/3<s<44/3<s<4 hold?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Geometric function theory, conformal mapping
Posed by
James E. Brennan, The integrability of the derivative in conformal mapping (J. London Math. Soc., 1978)
Year posed
1978
Years open
48y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every ε>0\varepsilon>0 there is CεC_\varepsilon with M−2[f′](r)≤Cε(1−r)−1−εM_{-2}[f'](r)\le C_\varepsilon(1-r)^{-1-\varepsilon} for all f∈Sf\in\mathcal S and 1/2≤r<11/2\le r<1; BS(−2)=1B_{\mathcal S}(-2)=1; and every conformal bijection φ:W→D\varphi:W\to\mathbb D has ∣φ′∣s|\varphi'|^s area-integrable for 4/3<s<44/3<s<4. Also BS(t)=∣t∣−1B_{\mathcal S}(t)=|t|-1 for all t≤−2t\le-2, and the Koebe function shows both endpoints fail. Not shown: the integral-means spectrum at other interior exponents; the companion's Bb(−1)<1/4B_b(-1)<1/4 is a separate entry.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The main theorem, its sharpness at both endpoints and the companion's strict inverse-first-power bound all have Lean formalizations in the release.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, Theorem 1.1 and the history section of 'Brennan's conjecture and sharp inverse-square integral means', read against Brennan's conjecture as recorded by Bertilsson. The proof (an operator on normalized half-plane maps, an eigen-law, and a critical-point pairing) was not refereed. Lean: formalization.yaml lists OAI.Brennan.main_theorem (lean/OAI/Analysis/IntegralMeans/Main.lean, comparator ComparatorChallenges/Brennan.lean). Its statement was read: a uniform bound M_{-2}[f'](r) <= C_eps (1-r)^(-1-eps) over the schlicht class, spectrum(-2) = 1, and for every open connected simply connected W with nontrivial spherical boundary and holomorphic bijection phi onto the disk, |phi'|^s integrable on W for 4/3 < s < 4, plus the disk form for -2 < t < 2/3. That is the headline. OAI.Brennan.Sharp.sharp_endpoints states divergence for the Koebe map at both endpoints. Permitted axioms propext, Quot.sound, Classical.choice. Not rebuilt here.

Sources

Changelog1 change

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