Brennan's conjecture and the integral-means spectrum

What OpenAI claims

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |ϕ'|^s is area-integrable for 4/3 < s < 4. The sharp universal integral-means identity is B_𝒮(t)=|t|-1 for t ≤ −2. A strict bound B_b(-1) < 1/4 for bounded univalent functions disproves Kraetzer's prediction at that parameter. (Lean)

Papers (2)

1. Brennan's conjecture and sharp inverse-square integral means

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then |φ'|^s is area-integrable for every 4/3 < s < 4. We also prove the sharp inverse-square integral-means exponent B_𝒮(-2)=1 for the normalized schlicht class 𝒮.

2. A strict inverse-first-power bound for univalent functions

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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 B_b(-1) < 1/4, disproving Kraetzer's conjectured spectrum at p = −1.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.