Analysis · Oct 6, 2026

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.

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