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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.

Manuscripts in this family · 2

Brennan's conjecture and sharp inverse-square integral means

Sep 24, 2026·paper.pdfLean

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

Full abstract

PDF on GitHub LaTeX source

A strict inverse-first-power bound for univalent functions

Sep 24, 2026·paper.pdfLean

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.

Full abstract

PDF on GitHub LaTeX source