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, $|\phi'|^s$ is area-integrable for $4/3\lt s\lt 4$. The sharp universal integral-means identity is $B_{\mathcal S}(t)=|t|-1$ for t ≤ −2. A strict bound $B_b(-1)\lt 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.
released 2026-09-24 | 1 theorem · 15 lemmas · 24 proofs · 11,485 words |
PLAY LEVEL 1 »(pdf)
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 $\mathcal S$.
released 2026-09-24 | 1 theorem · 9 lemmas · 15 proofs · 5,719 words |
PLAY LEVEL 2 »(pdf)
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 $B_b(-1)\lt 1/4$, disproving Kraetzer's conjectured spectrum at p = −1.