Brennan's conjecture (1978), a central open problem of geometric function theory, asserts that for every conformal (univalent) map ff of the unit disk the area integral ∬∣f′∣p dA\iint|f'|^p\,dA is finite for all p∈(−2,2/3)p\in(-2,2/3); equivalently, the universal integral means spectrum B(t)B(t) of univalent functions satisfies B(−2)=B^(−2)=1B(-2)=\hat B(-2)=1. Jin's study of a multiplier operator induced by the Schwarzian derivative of univalent functions verified the conjecture only for quasidisk classes controlled by the dilatation ∥μ∥∞≤5/8\|\mu\|_\infty\le\sqrt{5/8} or the boundary dilatation h(f)≤3/11h(f)\le\sqrt{3/11} of a quasiconformal extension; the problem here is the famous named conjecture itself, in its standard unrestricted form. No resolution is known as of mid-2026: the best unconditional integrability range is p∈(−p0,2/3)p\in(-p_0,2/3) with p0≈1.76p_0\approx1.76, and every recent advance proves only a special class (quasidisks, holomorphic semigroups, polynomial basins, univalent rational functions). The conjecture remains open.

Background

A univalent function in the unit disk D={z:∣z∣<1}\mathbb{D}=\{z:|z|<1\} is a holomorphic one-to-one map, that is, a conformal map of D\mathbb{D} onto some simply connected plane domain. Because the boundary behavior of such maps can be extremely irregular, a central quantitative probe is the integral means spectrum: for a real exponent tt one considers the integral means It(f,r)=∫02π∣f′(reiθ)∣t dθI_t(f,r)=\int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta and defines βf(t)\beta_f(t) as the infimum of the exponents γ\gamma such that It(f,r)≤C(f,γ)(1−r)−γI_t(f,r)\le C(f,\gamma)(1-r)^{-\gamma} for all r∈(0,1)r\in(0,1) — the polynomial growth rate of the tt-th means of f′f' as the circle ∣z∣=r|z|=r approaches the unit circle. The universal integral means spectrum is B(t)=sup⁡fβf(t)B(t)=\sup_f\beta_f(t), the supremum over all univalent ff in D\mathbb{D}, and B^(t)\hat B(t) is the same supremum restricted to bounded univalent functions. Makarov proved B(t)=max⁡{B^(t),3t−1}B(t)=\max\{\hat B(t),3t-1\}, and Carleson and Makarov established a threshold τ∗<0\tau^*<0 such that B(t)=B^(t)=∣t∣−1B(t)=\hat B(t)=|t|-1 for all t≤τ∗t\le\tau^*; currently only τ∗≤−2\tau^*\le-2 is known. These spectra govern the fine structure of harmonic measure (the boundary measure through which the Dirichlet problem is solved on a domain), so their exact values are a core problem of geometric function theory; the Hedenmalm–Sola survey gives a full account.

For t>0t>0 the means detect how large f′f' can be near the boundary; for t<0t<0 they detect how small it can be, that is, how strongly a conformal map can compress a neighborhood of a boundary point. Brennan's conjecture (1978) asserts B(−2)=B^(−2)=1B(-2)=\hat B(-2)=1 — equivalently τ∗=−2\tau^*=-2 — and in its original integral form asserts that

∬D∣f′(z)∣p dA(z)<∞\iint_{\mathbb{D}}|f'(z)|^p\,dA(z)<\infty

for every univalent ff in D\mathbb{D} and every p∈(−2,2/3)p\in(-2,2/3), where dAdA is planar Lebesgue measure. The upper endpoint 2/32/3 is sharp and not at issue: for the Koebe function f(z)=z/(1−z)2f(z)=z/(1-z)^2, which maps D\mathbb{D} onto the plane slit along a ray, the integral already diverges at p=2/3p=2/3. The entire content of the conjecture is the lower endpoint. It is known that the conjectured range holds for p∈(−p0,2/3)p\in(-p_0,2/3) with p0≈1.76p_0\approx1.76 (Sola), and the best current upper bound for B(−2)B(-2) is due to Hedenmalm and Shimorin, so the gap between the known endpoint ≈−1.76\approx-1.76 and the conjectured −2-2 has stood since 1978. Kraetzer's numerical experiments suggested the stronger law B^(t)=t2/4\hat B(t)=t^2/4 on [−2,2][-2,2] (with B^(t)=∣t∣−1\hat B(t)=|t|-1 outside), which contains Brennan's conjecture as its t=−2t=-2 instance and is likewise open.

A substantial literature verifies the conjecture only under auxiliary structure. A quasiconformal (QC) mapping is a homeomorphic Wloc1,2W^{1,2}_{loc} solution of the Beltrami equation ∂ˉf~=μf~ ∂f~\bar\partial\tilde f=\mu_{\tilde f}\,\partial\tilde f with ∥μf~∥∞<1\|\mu_{\tilde f}\|_\infty<1; its complex dilatation μ\mu measures the deviation from conformality, and a univalent ff admitting a QC extension to the Riemann sphere maps D\mathbb{D} onto a quasidisk. For such maps one can also use the boundary dilatation h(f)h(f), the infimum of ∥μ∥∞\|\mu\|_\infty over the exterior of compact subsets of the exterior disk, which isolates the dilatation contributed by the boundary alone. Building on Shimorin's multiplier estimate (a norm bound for multiplication by the Schwarzian derivative Sf=(f′′/f′)′−12(f′′/f′)2S_f=(f''/f')'-\frac12(f''/f')^2 acting on weighted Bergman spaces, the L2L^2 spaces of holomorphic functions with respect to weighted area measure), Hedenmalm obtained the conjecture for QC extensions with ∥μ∥∞≤0.523…\|\mu\|_\infty\le0.523\ldots, and Jin improved this to ∥μ∥∞≤5/8≈0.7906\|\mu\|_\infty\le\sqrt{5/8}\approx0.7906 and to h(f)≤3/11≈0.5222h(f)\le\sqrt{3/11}\approx0.5222, also proving the Kraetzer bound βf(−1)≤14\beta_f(-1)\le\frac14 for ∥μ∥∞≤1105/48≈0.6925\|\mu\|_\infty\le\sqrt{1105}/48\approx0.6925 or h(f)≤65/321≈0.4500h(f)\le\sqrt{65/321}\approx0.4500. These dilatation-controlled classes are genuinely special — a general univalent function need not admit any QC extension at all — which is exactly why they do not touch the general conjecture. Makarov also showed that it suffices to prove the conjecture for bounded univalent maps, equivalently (via Löwner chains, the differential-equation deformations of the identity that generate slit maps) for slit maps; this reduction has so far not been enough. Further partial confirmations come from complex dynamics (Barański–Volberg–Zdunik: the basins of infinity of the quadratic maps z↦z2+cz\mapsto z^2+c for parameters cc in the Mandelbrot set, the connectedness locus of the quadratic family; Zheng: basins of attraction of certain polynomials), from semigroup theory (Aleman–Kouroupis: continuous one-parameter semigroups of holomorphic self-maps of D\mathbb{D} under composition), and from numerics (Tošić: the known counterexample to Thurston's K=2K=2 conjecture is not a counterexample to Brennan's conjecture).

Problem Statement

The problem is Brennan's conjecture in its standard, unrestricted form. Prove or refute that

B(−2)=B^(−2)=1,B(-2)=\hat B(-2)=1,

that is, that for every univalent function ff in the unit disk D\mathbb{D},

∬D∣f′(z)∣p dA(z)<∞for every p∈(−2,2/3),\iint_{\mathbb{D}}|f'(z)|^p\,dA(z)<\infty\quad\text{for every }p\in(-2,2/3),

equivalently, that the Carleson–Makarov threshold satisfies τ∗=−2\tau^*=-2, i.e. the identity B(t)=B^(t)=∣t∣−1B(t)=\hat B(t)=|t|-1 extends to t=−2t=-2. The upper endpoint 2/32/3 is sharp and not in question (the Koebe function diverges there); the entire content is the lower endpoint −2-2. The statement must hold for arbitrary univalent maps — equivalently, for arbitrary simply connected plane domains — and no auxiliary hypothesis may be imposed: quasiconformal extensibility, a bound on the dilatation ∥μ∥∞\|\mu\|_\infty or the boundary dilatation h(f)h(f), membership in a holomorphic semigroup or in a structurally restricted subclass of Löwner chains, rationality, or a dynamical definition of the image domain each cut out strictly smaller derived problems, several of which are already settled (see the Background above). Makarov's reduction is of a different kind: proving the conjecture for bounded univalent maps — equivalently for slit maps and their Löwner chains — implies the conjecture in full generality, so a proof obtained through that reduction answers this problem rather than a strictly smaller one. The conjecture fails exactly when there exist a univalent ff in D\mathbb{D} and an exponent p∈(−2,2/3)p\in(-2,2/3) with ∬D∣f′∣p dA=∞\iint_{\mathbb{D}}|f'|^p\,dA=\infty. The stronger Kraetzer law B^(t)=t2/4\hat B(t)=t^2/4 on [−2,2][-2,2] is a neighboring open strengthening that implies this problem but is not part of its completion boundary.

References

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