Brennan's conjecture and sharp inverse-square integral means
We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then $|\varphi'|^s$ is area-integrable for every $4/3\lt s\lt 4$. We also prove the sharp inverse-square integral-means exponent $B_{\mathcal S}(-2)=1$ for the normalized schlicht class $\mathcal S$.
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Brennan's Conjecture and the Universal Integral Means SpectrumarXiv.orgIlia Binder, Oleg Ivrii, Istvan Prause, Stanislav SmirnovOct 16, 2026in the index
In October 2026, OpenAI released two preprints \cite{openai-brennan,openai-kraetzer}, establishing Brennan's celebrated conjecture~\cite{brennan,brennan2} and disproving Kraetzer's more general $t^2/4$ conjecture \cite{kraetzer, kraetzer-thesis}. While validating these results, we observed that…