Circular Brennan conjecture: the universal integral means spectrum on the circle $|t|=2$

The universal integral means spectrum $B(t)$ of conformal maps of the unit disk records the boundary growth of $\int_{|z|=r}|f'(z)^t|\,|dz|$ for complex exponents $t$, with the complex argument measuring rotational ('twisting') behavior. Brennan's classical conjecture asserts $B(-2)=1$; Becker and Pommerenke strengthened it radially to the circular Brennan conjecture, $B(t)=1$ for all complex $t$ with $|t|=2$, and Prause reformulated it in quasiconformally restricted form as $B_k(t)=1$ for $|t|=2/k$ for every quasiconformality parameter $k$, proving it on the arc $\operatorname{Re}t>2$. The conjecture — including its point $t=-2$, which is the classical Brennan conjecture open since 1978 — remains open as of mid-2026. This record states the source-faithful problem together with the partial results and identified obstacles.

Background

A conformal (univalent, i.e. holomorphic and injective) map $f$ from the unit disk $\mathbb D$ onto a bounded domain $\Omega$ compresses or expands boundary arcs near $\partial\mathbb D$ at rates encoded by its integral means spectrum

$$\beta_f(t)=\limsup_{r\to1}\frac{\log\int_{|z|=r}|f'(z)^t|\,|dz|}{\log\frac{1}{1-r}},\qquad t\in\mathbb C,$$

where the complex power is taken through the single-valued branch of $\log f'$ on $\mathbb D$ (available because $f'$ never vanishes there, with the branch fixed by $\arg f'(0)\in[0,2\pi)$). For real $t$, $\beta_f(t)$ measures boundary expansion ($t>0$) and compression ($t<0$); for complex $t$ it also measures rotation, i.e. twisting of the image boundary. Taking the supremum over all bounded conformal maps gives the universal integral means spectrum $B(t)$; restricting to maps that admit a $k$-quasiconformal extension to the whole plane — an extension to a homeomorphism of the plane whose complex dilatation satisfies $|\partial_{\bar z}f|\le k\,|\partial_z f|$ almost everywhere, $0\le k<1$ — gives $B_k(t)$. The image of $\mathbb D$ under such a global quasiconformal map is a quasidisk, the basic model domain of quasiconformal geometry, and a conformal map onto a quasidisk is exactly one with a quasiconformal extension.

Brennan's conjecture (1978), a standard open problem of geometric function theory, asserts $B(-2)=1$; equivalently, $\int_{\mathbb D}|f'|^p\,dA<\infty$ for every conformal $f$ and every $4/3<p<4$. The identity $B(2)=1$ is elementary (it follows from the finite area of $\Omega$), and the elementary lower bound $B(t)\ge|t|-1$ shows that the value $1$ cannot be improved anywhere on the circle $|t|=2$, so the conjecture is genuinely an upper-bound problem. Carleson and Makarov (1994) proved that the spectrum is exactly linear, $B(t)=|t|-1$, sufficiently far along the negative real axis, exhibiting a phase transition (between regimes where the supremum is dominated by isolated versus fractally distributed boundary singularities) that is conjectured to occur precisely at $t=-2$. Becker and Pommerenke (1987), in their study of the Hausdorff dimension of quasicircles, proposed the radial strengthening

$$B(t)=1\quad\text{for every complex }t\text{ with }|t|=2,$$

the circular Brennan conjecture. By the fractal approximation principle $B(t)=\sup_{k<1}B_k(t)$ (every conformal map is approximable by quasiconformally extendible ones), this is essentially equivalent to $B_k(t)=1$ on the circle $|t|=2/k$ for every $k$; the converse direction requires growth estimates for holomorphic motions (a family of injections of $\mathbb D$ depending holomorphically on a parameter $\lambda$; the $\lambda$-lemma guarantees such motions preserve conformality) that are uniform, i.e. independent of the particular motion — a stronger postulate already made by Becker and Pommerenke.

Prause (2019) proved the restricted identity $B_k(t)=1$ for $|t|=2/k$ with $\operatorname{Re}t>2$ (the real endpoint $B_k(2/k)=1$ is due to Prause and Smirnov, 2011), using a two-variable holomorphic motion and a three-point Nevanlinna–Pick interpolation problem on the bidisk (the Nevanlinna–Pick problem asks when a function taking prescribed values at prescribed points extends to a bounded analytic function; its three-point bidisk version supplies a two-variable maximum principle). His method leaves open the complementary arcs $\operatorname{Re}t\le2$ — including $t=-2$, the classical Brennan conjecture — and he records two limitations: the interpolation region $\mathcal{W}_k$ contains a second disk that the method cannot rule out as a 'non-physical' possibility, and passage from a priori bounds to the full circle needs motion-independent growth estimates. For unrestricted complex $t$ the best general bound is Hedenmalm's $B_k(t)\le(1+7k)^2k^2|t|^2/4$ (2017), which on the circle $|t|=2/k$ gives $(1+7k)^2>1$; Ivrii (2015) disproved the stronger conjecture $B_k(t)=k^2|t|^2/4$ for $|t|\le2/k$ by showing that quasicircles of Hausdorff dimension $1+k^2$ do not exist. Recent work of Jin (2024–2026) proves structural facts about these functionals for complex exponents — continuity on Teichmüller spaces (the parameter spaces of marked conformal structures on the disk) and the strict inequality $\beta_f(\tau)<B(\tau)$ for quasiconformally extendible $f$ — and verifies the classical Brennan conjecture for special classes (rational functions, polygonal images), but the circle identity itself is untouched.

Problem Statement

Prove or disprove the circular Brennan conjecture of Becker and Pommerenke (recorded as Conjecture 1.1 in Prause 2019): the universal integral means spectrum of conformal maps satisfies

$$B(t)=1\quad\text{for every complex }t\text{ with }|t|=2.$$

In the quasiconformally restricted form conjectured in display (1.4) of that paper, prove or disprove that

$$B_k(t)=1\quad\text{for every }0<k<1\text{ and every }t\text{ with }|t|=\tfrac{2}{k},$$

with no restriction on $\operatorname{Re}t$ (Prause's Theorem 1.2 covers only the arc $\operatorname{Re}t>2$; the fractal approximation principle $B(t)=\sup_{k<1}B_k(t)$ makes the two formulations essentially equivalent, the converse direction requiring uniform, motion-independent growth estimates for holomorphic motions). Since the lower bounds $B(t)\ge|t|-1$ and $B_k(t)\ge k|t|-1$ are elementary and equal to $1$ on the respective circles, the open content is the upper bound $B(t)\le1$ (resp. $B_k(t)\le1$) on the full circle; the point $t=-2$ is the classical Brennan conjecture.

References

1. István Prause, Quasidisks and twisting of the Riemann map, Int. Math. Res. Not. IMRN 2019, no. 7, 1936–1954. DOI: 10.1093/imrn/rnx173; arXiv:1611.06734. https://arxiv.org/abs/1611.06734
2. J. Becker, C. Pommerenke, On the Hausdorff dimension of quasicircles, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 329–333. DOI: 10.5186/aasfm.1987.1206. https://doi.org/10.5186/aasfm.1987.1206
3. James E. Brennan, The integrability of the derivative in conformal mapping, J. London Math. Soc. (2) 18 (1978), no. 2, 261–272. DOI: 10.1112/jlms/s2-18.2.261. https://doi.org/10.1112/jlms/s2-18.2.261
4. István Prause, Stanislav Smirnov, Quasisymmetric distortion spectrum, Bull. Lond. Math. Soc. 43 (2011), no. 2, 267–277. DOI: 10.1112/blms/bdq098. https://doi.org/10.1112/blms/bdq098
5. Haakan Hedenmalm, Bloch functions and asymptotic tail variance, Adv. Math. 313 (2017), 947–990. DOI: 10.1016/j.aim.2017.04.016. https://doi.org/10.1016/j.aim.2017.04.016
6. Håkan Hedenmalm, Serguei Shimorin, Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J. 127 (2005), no. 2, 341–393. DOI: 10.1215/s0012-7094-04-12725-3. https://doi.org/10.1215/s0012-7094-04-12725-3
7. Lennart Carleson, Nikolai G. Makarov, Some results connected with Brennan's conjecture, Ark. Mat. 32 (1994), no. 1, 33–62. DOI: 10.1007/BF02559522. https://doi.org/10.1007/BF02559522
8. Oleg Ivrii, Quasicircles of dimension 1+k^2 do not exist, arXiv:1511.07240 (2015). https://arxiv.org/abs/1511.07240
9. Haakan Hedenmalm, Quasicircles and hyperbolic zero packing, arXiv:2207.09846 (2022). https://arxiv.org/abs/2207.09846
10. Jianjun Jin, Complex exponential integral means spectra of univalent functions and the Brennan conjecture, arXiv:2512.09330 (2025; latest revision 5 June 2026). https://arxiv.org/abs/2512.09330
11. Jianjun Jin, Integral means spectrum functionals on Teichmüller spaces, arXiv:2405.07683 (2024). https://arxiv.org/abs/2405.07683
12. Jianjun Jin, On the integral means spectrum of univalent functions with quasconformal extensions, arXiv:2407.19240 (2024). https://arxiv.org/abs/2407.19240
13. Alexandru Aleman, Athanasios Kouroupis, Brennan's conjecture holds for semigroups of holomorphic functions, arXiv:2409.15074 (2024). https://arxiv.org/abs/2409.15074
14. Yigang Zheng, Brennan Conjecture for Basin of Attraction at Infinity, arXiv:2604.12240 (2026). https://arxiv.org/abs/2604.12240
15. Olli Hirviniemi, István Prause, Eero Saksman, Stretching and Rotation of Planar Quasiconformal Mappings on a Line, arXiv:2007.07735 (2020). https://arxiv.org/abs/2007.07735
16. Olli Hirviniemi, Lauri Hitruhin, István Prause, Eero Saksman, On mappings of finite distortion that are quasiconformal in the unit disk, arXiv:2104.10961 (2021). https://arxiv.org/abs/2104.10961