OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |\phi'|^s is area-integrable for 4/3< s< 4. The sharp universal integral-means identity is B_{\mathcal 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.
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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< s< 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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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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
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