Brennan's conjecture and the integral-means spectrum

1 · The essentials

An angle-preserving map can turn a region of the plane into a disk, but its local stretching can become very large or very small. Brennan's conjecture asks whether certain powers of that stretching still have a finite total when added up over the original region. The main manuscript establishes this for every power strictly between $4/3$ and $4$ , provided the region has no holes and the map matches its points one-to-one with the disk. It also gives an exact limit on a related measure of stretching for another specified collection of maps. A companion manuscript gives a stricter limit for a different measure, contradicting a prediction attributed to Kraetzer at one particular value. Both manuscripts are listed as Lean-formalized, although the supplied descriptions do not specify the full coverage of those formalizations.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine reshaping a patterned sheet into a circular sheet while keeping the angles of the pattern intact. The question is not whether every tiny patch stretches by a bounded amount, but whether a chosen power of those stretching factors has a finite total over the sheet. This image illustrates the question, not the mathematical argument.

Why it matters

The main result settles Brennan's conjecture under the hypotheses and within the open range of powers stated in the source. The accompanying spectrum results distinguish an exact universal value from a strict upper bound. In particular, the strict bound contradicts Kraetzer's predicted spectrum at the parameter $- 1$ , rather than merely providing another estimate.

2 · Going further

The precise integrability problem concerns a conformal bijection $φ : Ω \to D$ , where $Ω$ is a simply connected plane domain and $D$ is the unit disk. The main manuscript establishes that $\int_{Ω} ∣ φ^{'} (z) ∣^{s} d A (z) < \infty$ whenever $4/3 < s < 4$ . Here $∣ φ^{'} ∣$ measures local stretching, and $d A$ denotes integration with respect to area. The assumptions include both simple connectedness and the existence of the stated conformal bijection.

The same manuscript establishes the sharp inverse-square integral-means exponent $B_{S} (- 2) = 1$ for the normalized schlicht class $S$ . The repository summary additionally states the universal identity $B_{S} (t) = ∣ t ∣ - 1$ for $t \leq - 2$ . Thus the value at $- 2$ appearing in the main abstract is also the endpoint value of the broader identity reported in the summary. The supplied abstracts announce these conclusions but do not describe the argument or explain how the integrability result and the spectrum identity are connected.

The companion manuscript addresses inverse-first-power integral means of normalized univalent disk maps. It establishes a uniform upper bound with exponent strictly below $1/4$ , with the stated consequence $B_{b} (- 1) < 1/4$ for the bounded universal integral-means spectrum. The source identifies this as a contradiction to Kraetzer's conjectured spectrum at $p = - 1$ . Both manuscripts are marked Lean-formalized; the supplied material does not give a theorem-by-theorem account of what the formalizations cover.

Useful notions

Complex derivatives and local stretching
Conformal bijections
Simply connected plane domains
Area integration and integrability
Univalent functions and integral means

What was known before / what is new

Before

The source identifies the area-integrability question as Brennan's conjecture. It also identifies a conjectured spectrum attributed to Kraetzer, which the companion result contradicts at $p = - 1$ . No earlier estimates, chronology, or history of the arguments are supplied.

New

The main manuscript establishes Brennan's integrability statement throughout $4/3 < s < 4$ and the sharp value $B_{S} (- 2) = 1$ . The repository summary reports the broader identity $B_{S} (t) = ∣ t ∣ - 1$ for $t \leq - 2$ . The companion manuscript establishes a uniform inverse-first-power estimate yielding $B_{b} (- 1) < 1/4$ .

Limits and assumptions

The integrability result assumes a simply connected plane domain admitting a conformal bijection onto the unit disk, and it covers only the strict range $4/3 < s < 4$ ; the source gives no endpoint conclusions. The reported universal spectrum identity is restricted to $t \leq - 2$ , while the strict bounded-spectrum inequality concerns the parameter $- 1$ . Both manuscripts are listed as Lean-formalized, but the supplied text does not specify the precise formalization scope, particularly for the broader identity appearing only in the repository summary. The abstracts provide no proof strategy and do not identify what remains open outside their stated conclusions.

Mini glossary

Conformal bijection: A one-to-one, onto map that preserves local angles.
Simply connected plane domain: A connected open region of the plane with no holes.
Area-integrable: Having a finite integral over the region with respect to area.
Univalent function: A one-to-one holomorphic function.
Integral-means spectrum: A function recording growth exponents for integral means at different power parameters.

Keywords : Brennan's conjecture · Integral-means spectrum · Univalent functions

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $∣ ϕ^{'} ∣^{s}$ is area-integrable for $4/3 < s < 4$ . The sharp universal integral-means identity is $B_{S} (t) = ∣ t ∣ - 1$ for t ≤ −2. A strict bound $B_{b} (- 1) < 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

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lean/docs/072.md

Manuscripts in this family

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Brennan's conjecture and sharp inverse-square integral means

24/09/2026

Verified by Lean

We prove Brennan's conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then $∣ φ^{'} ∣^{s}$ is area-integrable for every $4/3 < s < 4$ . We also prove the sharp inverse-square integral-means exponent $B_{S} (- 2) = 1$ for the normalized schlicht class $S$ .

PDF GitHub folder Lean formalization

A strict inverse-first-power bound for univalent functions

24/09/2026

Verified by Lean

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.

PDF GitHub folder Lean formalization