Kraetzer's conjecture on the universal integral means spectrum

For a bounded univalent map $f$ of the unit disk, $\beta_f(p)$ is the growth exponent of $\int|f'(re^{i\theta})|^p d\theta$ as $r\to1$ , and the universal bounded spectrum is $B_b(p)=\sup_f\beta_f(p)$ . From numerical experiments Kraetzer (1996) conjectured $B_b(p)=p^2/4$ for $|p|\le2$ and $B_b(p)=|p|-1$ for $|p|\ge2$ . The conjecture contains Brennan's conjecture ( $B(-2)=1$ ) and the value $B_b(1)=1/4$ related to the Carleson-Jones coefficient problem. Hedenmalm-Shimorin and Sola gave upper bounds above $1/4$ at $p=-1$ . Is Kraetzer's formula for $B_b$ correct?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Geometric function theory, integral means spectrum
Posed by: Philipp Kraetzer, Experimental bounds for the universal integral means spectrum of conformal maps (Complex Variables, 1996)
Year posed: 1996
Years open: 30y
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: 25 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: there are constants $0<\varepsilon<1/4$ and $C$ with $M_{-1}[f'](r)\le C(1-r)^{-1/4+\varepsilon}$ for every $f\in\mathcal S$ and $1/2\le r<1$ , so $B_b(-1)<1/4=p^2/4$ at $p=-1$ and Kraetzer's formula is false. The bound needs no bounded image or boundary regularity. Not shown: the true value of $B_b(-1)$ or any explicit gap, and nothing about $p=1$ or the Carleson-Jones question; Brennan's value $B(-2)=1$ , also predicted by Kraetzer, is proved in the companion (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. This is the companion of the Brennan manuscript in the same family; the two share the affine-law and critical-point method but the paper says neither is an input to the other. It has a Lean formalization in the release.

Verification

No independent mathematician has checked this yet. Checked here: the abstract and Theorem 1.1 of 'A strict inverse-first-power bound for univalent functions', read against Kraetzer's formula as the paper quotes it (1996 paper, 2000 thesis, Beliaev's restatement). The proof was not refereed. Lean: formalization.yaml lists OAI.StrictInverseFirstPower.main (lean/OAI/Analysis/StrictMeans/Main.lean, comparator ComparatorChallenges/StrictMeans.lean). Its statement was read: there are 0 < eps < 1/4 and C with M_{-1}[f'](r) <= C (1-r)^(-1/4+eps) for every normalized univalent f, the bounded spectrum at -1 is below 1/4, and it is not the case that boundedSpectrum p equals the piecewise Kraetzer prediction for all p. That is the headline. Not rebuilt here. The gap eps is existential; no numerical value is given.

Kraetzer's conjecture on the universal integral means spectrum

What was actually shown

What the AI did

Verification

Sources

Discussion