Result 072, Real and complex analysis

Brennan's conjecture and the integral-means spectrum

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, ∣ϕ′∣s|\phi'|^s is area-integrable for 4/3<s<44/3\lt s\lt 4. The sharp universal integral-means identity is BS(t)=∣t∣−1B_{\mathcal S}(t)=|t|-1 for t ≤ −2. A strict bound Bb(−1)<1/4B_b(-1)\lt 1/4 for bounded univalent functions disproves Kraetzer's prediction at that parameter.

Lean formalization Proof

The bigger picture

Why it matters

Conformal maps preserve angles but can stretch distances dramatically near a boundary. These manuscripts claim sharp limits on that distortion, together with a bound that contradicts a proposed universal growth law.

What changes?

The manuscript reports that every conformal bijection from a simply connected plane domain onto the unit disk has an area-integrable derivative magnitude raised to any power strictly between 4/3 and 4. For normalized univalent disk maps, meaning injective holomorphic maps with a fixed normalization, the reported universal integral-means spectrum equals the absolute value of t minus 1 for all t at most -2. In particular, the inverse-square exponent is exactly 1.

What does that help mathematicians do?

Integral-means spectra measure how quickly circular averages of powers of a map's derivative can grow as the circle approaches the disk boundary. The second manuscript reports a uniform inverse-first-power growth exponent strictly below 1/4 for normalized univalent disk maps. Consequently, the spectrum for bounded maps is below 1/4 at parameter -1. This rules out Kraetzer's proposed spectrum there: the prediction overestimates the worst possible growth, rather than merely missing an exceptional example.

Are there practical applications?

The immediate value is foundational: the claimed integrability theorem would let researchers control area integrals of powers of conformal stretching throughout the stated range, without additional boundary assumptions. This is control of accumulated distortion, not a pointwise cap on stretching. The supplied material establishes no practical algorithm or deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Brennan's conjecture and sharp inverse-square integral means

September 24, 2026 31 pages Main result formalized in Lean

We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then ∣φ′∣s|\varphi'|^s is area-integrable for every 4/3<s<44/3\lt s\lt 4. We also prove the sharp inverse-square integral-means exponent BS(−2)=1B_{\mathcal S}(-2)=1 for the normalized schlicht class S\mathcal S.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026,
  author = {{OpenAI}},
  title = {{Brennan's conjecture and sharp inverse-square integral means}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf}{OAI:Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026}},
  year = {2026}
}

A strict inverse-first-power bound for univalent functions

September 24, 2026 15 pages Main result formalized in Lean

We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4. Consequently, the bounded universal integral-means spectrum satisfies Bb(−1)<1/4B_b(-1)\lt 1/4, disproving Kraetzer's conjectured spectrum at p = −1.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026,
  author = {{OpenAI}},
  title = {{A strict inverse-first-power bound for univalent functions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf}{OAI:A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/072.md.

Brennan's conjecture and the integral-means spectrum

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Brennan's conjecture asserts that ∣ϕ′∣s|\phi'|^s is area-integrable for 4/3<s<44/3<s<4 when ϕ\phi conformally maps a simply connected plane domain with nontrivial spherical boundary onto the disk. The formalization establishes this interval, along with area integrability of ∣f′∣t|f'|^t for univalent disk maps when −2<t<2/3-2<t<2/3.

It also gives the uniform inverse-square radial-mean bound with exponent −1−ε-1-\varepsilon for every ε>0\varepsilon>0 and the spectrum value BS(−2)=1B_{\mathcal S}(-2)=1. The Koebe map and its inverse give divergence at all four boundary exponents.

Kraetzer's proposed integral-means spectrum predicts Bb(−1)=1/4B_b(-1)=1/4 for bounded univalent maps. The formalized result proves Bb(−1)<1/4B_b(-1)<1/4, contradicting that prediction.

The underlying estimate gives constants 0<ε<1/40<\varepsilon<1/4 and C<∞C<\infty such that the normalized circular mean of ∣f′∣−1|f'|^{-1} is at most C(1−r)−1/4+εC(1-r)^{-1/4+\varepsilon} for every normalized univalent disk map and 1/2≤r<11/2\le r<1. The bound requires neither bounded image nor boundary regularity, and no numerical value of ε\varepsilon is specified.

Comparator links

Result Comparator statement
Brennan and inverse-square integral-means bounds Brennan.lean
Sharp endpoint divergence BrennanSharp.lean
Strict inverse-first-power bound and consequences StrictMeans.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 11 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.