OpenAI · 072-01 · 2026-09-24 · Real and complex analysis
Brennan's conjecture and sharp inverse-square integral means
preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026
Family 072. Brennan's conjecture and a counterexample to Kraetzer's prediction.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves Brennan's conjecture: for every conformal bijection from a simply connected plane domain onto the disk, is area-integrable for . The sharp universal integral-means identity is for . A strict bound for bounded univalent functions disproves Kraetzer's prediction at that parameter.
We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then is area-integrable for every . We also prove the sharp inverse-square integral-means exponent for the normalized schlicht class .
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 072: Brennan's conjecture and the integral-means spectrum. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.