Brennan's Conjecture and the Universal Integral Means Spectrum

Abstract: In October 2026, OpenAI released two preprints \cite{openai-brennan,openai-kraetzer}, establishing Brennan's celebrated conjecture~\cite{brennan,brennan2} and disproving Kraetzer's more general $t<sup>2/4$ conjecture \cite{kraetzer, kraetzer-thesis}. While validating these results, we observed that much of their argument is redundant and arrived at a drastically shorter proof of a significantly stronger statement. The goal of this note is to give a short and concise presentation of the reborn proof, sacrificing some of the context and consequences for brevity, and relegating them to a follow-up work. In this particular case the human-AI interaction was fruitful and quite satisfactory, and hoping for a similar future in other subjects, we would paraphrase Rick Blaine: {\em ``AI, we hope this is the beginning of a beautiful friendship.''}

• The paper proves that the bounded universal integral means spectrum satisfies \(\mathcal B_b(t)\le t^2/4\) for \(-2\le t\le2\), using superharmonic potentials, critical-point counting, signed-area estimates, and weighted integral means.
• Brennan’s conjecture follows at \(t=-2\), yielding conformal-derivative integrability for every exponent \(3<p<4\), while Makarov’s identity extends the improvement to the unrestricted spectrum on \((-2,6-4\sqrt2)\).
• The paper shows that Kraetzer’s proposed equality \(\mathcal B_b(t)=t^2/4\) cannot hold throughout the interior, although the detailed proof of strictness and the characterization of extremal maps are deferred or remain open.

Problem setting and principal result

The paper studies the universal integral means spectrum of conformal maps and derives a strengthened form of the bounds associated with Brennan’s conjecture. For a univalent map $f:\mathbb D\to\mathbb C$, the integral means spectrum is defined by

$$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$

The bounded and unrestricted universal spectra are respectively

$$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$

where the first supremum is over bounded univalent maps and the second over all univalent maps. The paper’s central theorem is

$$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$

with strict inequality for $t\in(-2,2)\setminus\{0\}$. Via Makarov’s identity,

$\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$

the authors obtain

$\mathcal B(t)<\frac{t^2}{4}$

for

$t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$

The endpoint $t=-2$ corresponds to Brennan’s conjecture. Under the inverse-map change of variables, the assertion $\mathcal B(-2)=1$ yields the integrability range

$$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$0

for conformal maps $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$1. Thus the paper proves not merely the endpoint estimate required by Brennan’s conjecture but a strict sub-Kraetzerian upper bound for the bounded spectrum throughout the interior of the interval $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$2.

The result also contradicts the proposed exact formula $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$3 in the interior, except at $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$4 and potentially at the endpoints. The paper therefore separates the validity of the quadratic expression as an upper bound from its invalidity as a universal sharp formula.

Relation to previous conjectures

Brennan’s conjecture predicts that conformal derivatives possess area-integrability up to, but not including, exponent $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$5. In the integral-means formulation, this is equivalent to the assertion $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$6. The geometric model motivating the conjecture is an inward-pointing slit, where the inverse conformal map has derivative comparable to $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$7 near the slit tip. The conjecture states that configurations involving multiple slits cannot produce a worse singularity after their mutual screening is taken into account.

Kraetzer proposed the stronger quadratic spectrum

$$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$8

together with $$\beta_f(t)=\limsup_{r\uparrow 1} \frac{\log \int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta} {\log(1/(1-r))}.$$9 outside this interval. The present paper establishes the non-strict part of the quadratic upper bound and states that the inequality is strict away from $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$0. Consequently, Kraetzer’s formula cannot be correct as an equality on the interior. The paper emphasizes that the strict result is qualitatively stronger than Brennan’s conjecture, because Brennan only probes the single value $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$1.

The potential-theoretic construction

The proof begins with a bounded univalent function $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$2 that is analytic across the closed disk. This regularity is obtained by first replacing a general bounded map $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$3 with the dilated map

$$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$4

and then letting $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$5. The estimates are uniform in $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$6, so the dilation does not alter the resulting spectrum bound.

For $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$7, the authors introduce the potential

$$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$8

where $$\mathcal B_b(t)=\sup_f\beta_f(t), \qquad \mathcal B(t)=\sup_f\beta_f(t),$$9 and $$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$0. Writing $$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$1, the logarithm

$$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$2

satisfies

$$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$3

away from a possible pole satisfying $$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$4. Hence $$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$5 is strictly superharmonic. Since $$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$6 tends to zero at the unit circle, its positive superlevel sets are relatively compact in the disk. This compactness permits the use of critical-point and index arguments on the superlevel components.

The auxiliary map

$$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$7

encodes the parameter $$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$8 for which $$\mathcal B_b(t)\le \frac{t^2}{4}, \qquad -2\le t\le 2,$$9 is a critical point of $t\in(-2,2)\setminus\{0\}$0. Specifically, away from the origin and a possible pole,

$t\in(-2,2)\setminus\{0\}$1

At such a critical point, the potential height is related to

$t\in(-2,2)\setminus\{0\}$2

through

$t\in(-2,2)\setminus\{0\}$3

The Jacobian determinant of $t\in(-2,2)\setminus\{0\}$4 determines the Morse type of the critical point. Because $t\in(-2,2)\setminus\{0\}$5 is superharmonic, a positive Jacobian corresponds to a saddle and a negative Jacobian to a local maximum. This correspondence is the mechanism that converts the topology of superlevel sets into a signed-area inequality.

Critical-point counting and the signed-area inequality

For a regular height $t\in(-2,2)\setminus\{0\}$6, each component of $t\in(-2,2)\setminus\{0\}$7 is a Jordan domain. A hole would force an interior minimum of $t\in(-2,2)\setminus\{0\}$8, which is impossible under strict superharmonicity. Applying the index theorem to each component gives

$t\in(-2,2)\setminus\{0\}$9

where $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$0, $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$1, and $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$2 count maxima, saddles, and poles, respectively. Since univalence permits at most one pole, summing over all superlevel components yields

$\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$3

Integrating this inequality over the height parameter and using the relation between critical heights and $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$4 gives a global signed-area estimate. Applying the real area formula separately to the regions where $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$5 is positive and negative produces

$\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$6

for suitable nonnegative test functions $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$7. Equivalently, the positive-Jacobian contribution, associated with saddles, cannot dominate the negative-Jacobian contribution, associated with maxima, after weighting by $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$8.

This inequality is the central geometric input. It is not an estimate obtained directly from pointwise control of $\mathcal B(t)=\max\{\mathcal B_b(t),3t-1\},$9; rather, it arises from a global comparison between the critical-point geometry of a family of superharmonic potentials and the area distortion of the map $\mathcal B(t)<\frac{t^2}{4}$0.

Conversion to integral means

The proof next expresses the signed-area integral in terms of the integral means

$\mathcal B(t)<\frac{t^2}{4}$1

Differentiating in polar coordinates gives

$\mathcal B(t)<\frac{t^2}{4}$2

and

$\mathcal B(t)<\frac{t^2}{4}$3

The authors define a carefully chosen differential expression $\mathcal B(t)<\frac{t^2}{4}$4 involving $\mathcal B(t)<\frac{t^2}{4}$5 and $\mathcal B(t)<\frac{t^2}{4}$6, with parameters

$\mathcal B(t)<\frac{t^2}{4}$7

A direct calculation transforms the signed-area density into

$\mathcal B(t)<\frac{t^2}{4}$8

This identity is important because all interior terms involving $\mathcal B(t)<\frac{t^2}{4}$9 are absorbed into the boundary expression $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$0. The remaining bulk term has a definite sign sufficiently close to the boundary: if $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$1 is chosen so that $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$2, then

$t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$3

After integrating radially and controlling the signed-area contribution through the previously established inequality, the authors obtain the weighted estimate

$t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$4

uniformly over the dilation parameter $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$5.

The boundary correction is controlled using boundedness of $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$6 and the fact that the image of the outer annulus under $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$7 remains in a fixed compact region separated from the singular behavior of $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$8 near the origin. The condition $t\in\left(-2,\,6-4\sqrt 2\right)\setminus\{0\}.$9 ensures integrability of the weight at the boundary. Monotone convergence then transfers the estimate from the dilated maps to the original bounded map.

Since $t=-2$0 is nondecreasing in $t=-2$1, the weighted integral estimate implies

$t=-2$2

and therefore

$t=-2$3

Letting $t=-2$4 gives

$t=-2$5

This proves the claimed upper bound for every $t=-2$6, $t=-2$7. The case $t=-2$8 is immediate, while $t=-2$9 follows from the area identity for bounded univalent maps and the monotonicity of $\mathcal B(-2)=1$0.

Strictness and the strength of the theorem

The displayed proof in the paper establishes the non-strict inequality directly. Strictness is obtained by perturbing the potential with an additional factor involving the pre-Schwarzian ratio $\mathcal B(-2)=1$1 and then replacing pointwise positivity by an averaged positivity argument. The paper states that this yields

$\mathcal B(-2)=1$2

but explicitly defers the details to a separate work.

This distinction matters. The non-strict estimate is fully developed through the potential, index, area, and integral-means argument presented in the paper. The strict estimate is asserted as a consequence of a modification whose technical implementation is not included in the current proof. Accordingly, the paper’s strongest stated theorem depends, for its strict component, on an argument that is described rather than completely documented here.

The resulting unrestricted-spectrum estimate follows from Makarov’s identity. The inequality

$\mathcal B(-2)=1$3

holds on the relevant interval up to $\mathcal B(-2)=1$4, so the bounded-spectrum estimate controls the unrestricted spectrum there. The paper therefore improves the quadratic upper bound for $\mathcal B(-2)=1$5 on a nontrivial interval containing negative exponents, precisely the region most directly connected with inverse derivative integrability and Brennan’s conjecture.

Limitations and open questions

The proof assumes analytic extension across the unit circle during its main estimates and removes this assumption by dilation and a limiting argument. This reduction is standard within the framework of integral means, but it requires uniformity in the dilation parameter; the paper’s argument depends on that uniformity at several points, particularly in the control of the image under $\mathcal B(-2)=1$6 and the weighted radial integral.

The strict inequality is not proved in full detail in the presented text. The perturbation of the potential is specified, but the averaged positivity argument is deferred. Thus the strict theorem is not supported by the same level of explicit derivation as the non-strict bound.

The method also does not identify extremal maps. The authors state that the argument can yield stronger numerical estimates, but it is unclear whether it can produce a sharp upper estimate or characterize maps attaining the spectrum. In particular, the proof establishes a structural obstruction to the Kraetzer equality without determining the actual universal spectrum. The paper leaves open whether the signed-area method can be optimized to obtain the exact spectrum or whether a different mechanism is needed to describe extremizers.

Conclusion

The paper develops a short potential-theoretic proof of the quadratic upper bound

$\mathcal B(-2)=1$7

for the bounded universal integral means spectrum on $\mathcal B(-2)=1$8. Its central mechanism combines superharmonic potentials, critical-point index counting, the Jacobian geometry of an auxiliary map, and a signed-area inequality that is converted into a weighted integral-means estimate. The result implies Brennan’s conjecture and, more strongly, rules out the Kraetzer spectrum as an equality throughout the interior of the critical interval. The non-strict estimate is established by the argument given, while the strict improvement and the identification of extremal behavior remain incompletely resolved within the paper (2610.23853).