Brennan's integrability conjecture for the basin of infinity of general monic polynomials

For a monic complex polynomial $f(z)=z^m+a_{m-1}z^{m-1}+\cdots+a_0$ of degree $m\ge3$ whose basin of attraction of infinity $\Omega(f)$ is simply connected (equivalently, whose filled Julia set is connected), the problem is to prove — or disprove by an explicit example — that Brennan's integrability conjecture holds for this dynamically defined domain: for a conformal map $F:\Omega(f)\to\mathbb{D}$ onto the unit disk, $\int_{\Omega(f)}|F'|^p\,dA<\infty$ for every $p\in(4/3,4)$, with no smallness restriction on the non-leading coefficients. The conjecture is known in this setting only for the quadratic family (Barański–Volberg–Zdunik 1998), for the unicritical family $z^m+a_0$, and for perturbations with $|a_k|<\delta(m)$ for an unspecified small threshold (Zheng 2026).

Background

Brennan's conjecture is a classical open problem on the $L^p$-integrability of derivatives of conformal maps. If $\Omega\subsetneq\mathbb{C}$ is a simply connected domain and $F:\Omega\to\mathbb{D}$ is a conformal map (a biholomorphism onto the unit disk, which exists by the Riemann mapping theorem), J. E. Brennan conjectured in 1978 that $\int_\Omega|F'|^p\,dA<\infty$ for every $p\in(4/3,4)$, where $dA$ is planar area measure. The case $p=2$ is trivial (the integral computes the area of the disk), while both endpoints are sharp: for the complement of a slit the integral diverges at $p=4/3$ and at $p=4$. The conjecture is open in general; the best published universal range extends beyond $p=3.7$ (Hedenmalm and Shimorin proved convergence for $4/3<p<3.75$, improving earlier work of Bertilsson, as surveyed in Zheng's 2026 introduction), and no counterexample is known for any simply connected domain.

This problem concerns a dynamically defined class of domains. For a polynomial $f$ of degree $m\ge2$, the basin of attraction of infinity is $\Omega(f)=\{z\in\hat{\mathbb{C}}: f^n(z)\to\infty\}$, where $f^n$ denotes the $n$-fold iterate; its complement is the filled Julia set $K(f)$, the set of points with bounded forward orbit, whose boundary is the Julia set. A planar domain is simply connected exactly when its complement in the Riemann sphere is connected, so $\Omega(f)$ is simply connected if and only if $K(f)$ is connected, which for polynomials holds if and only if no finite critical point of $f$ escapes to infinity (a classical criterion in polynomial dynamics). On such a basin the Riemann map $F:\Omega(f)\to\mathbb{D}$ exists, and if $R=F^{-1}$ is normalized by $R(0)=\infty$ then the conjugate $R^{-1}\circ f\circ R$ must be of the form $z\mapsto az^m$ with $|a|=1$, since the ratio $g(z)/z^m$ is holomorphic and nonvanishing on $\mathbb{D}$ with boundary values of modulus one. One may therefore ask whether Brennan's integrability holds for this $F$.

Barański, Volberg, and Zdunik (1998) proved that it does for every quadratic polynomial $f(z)=z^2+a_0$ with simply connected basin. Zheng (2026) extended this to the unicritical family $f(z)=z^m+a_0$ for all $m\ge3$ — where the only finite critical point is $0$ and rotational symmetry is available — and, by a perturbation argument, to every monic $f(z)=z^m+\sum_{k=0}^{m-1}a_kz^k$ whose non-leading coefficients satisfy $|a_k|<\delta(m)$ for $1\le k\le m-1$, for some unspecified threshold $\delta(m)>0$ (the constant term $a_0$ is unconstrained). The proof follows the Barański–Volberg–Zdunik strategy: by Koebe's distortion theorem (standard two-sided bounds on the derivative of a univalent function relative to its value at a base point), convergence of $\int_{\Omega(f)}|F'|^p$ for $2\le p\le4$ is reduced to convergence of the series $\sum_n S_n/m^{4n}$, where $S_n=\sum_{k=0}^{m^n-1}|(f^n)'(w_{n,k})|^2$ sums squared derivatives along the backward orbit of a base point $w_0$; a sufficient criterion is $\limsup_{n\to\infty}S_{n+1}/S_n<m^4$. For $f(z)=z^m+a_0$ rotational symmetry yields the closed-form bound $\limsup S_{n+1}/S_n\le m^3\,2^{(m+1)/m}<m^4$, and the perturbation theorem compares preimage sums of $f$ with those of $z^m+a_0$ via continuity of polynomial roots — but that comparison is uniform only on compact sets, so it works only while the lower coefficients stay below the threshold $\delta(m)$. Neither the size nor the behavior of $\delta(m)$ is determined, and monic polynomials with large lower-order coefficients (equivalently, with several critical points at arbitrary dynamically relevant positions) are not covered. For the unicritical basins Zheng's argument in fact gives integrability slightly beyond $p=4$, so the optimal exponent range for polynomial basins is itself unsettled.

Problem Statement

Let $m\ge3$ and let $f(z)=z^m+a_{m-1}z^{m-1}+\cdots+a_1z+a_0$ be a monic complex polynomial whose basin of attraction of infinity

$\Omega(f)=\{z\in\hat{\mathbb{C}}: f^n(z)\to\infty\}$

is simply connected — equivalently, whose filled Julia set is connected, i.e. all finite critical points of $f$ have bounded forward orbits — with no restriction on the sizes of the coefficients $a_0,\dots,a_{m-1}$. Let $F:\Omega(f)\to\mathbb{D}$ be a conformal map onto the unit disk. Prove, or disprove by an explicit polynomial, that

$\int_{\Omega(f)}|F'(z)|^p\,dA(z)<\infty\quad\text{for every }p\in(4/3,4).$

That is, establish Brennan's integrability conjecture for the basin of infinity of every monic polynomial of degree $m\ge3$ with connected filled Julia set, removing the smallness condition $|a_k|<\delta(m)$ on non-leading coefficients in Zheng's perturbative theorem. A refutation consists of one explicit monic polynomial of degree $\ge3$ with simply connected basin and one $p\in(4/3,4)$ for which the integral provably diverges.

References

1. Yigang Zheng, "Brennan Conjecture for Basin of Attraction at Infinity", arXiv preprint arXiv:2604.12240 (2026), https://arxiv.org/abs/2604.12240
2. Krzysztof Barański, Alexander Volberg, Anna Zdunik, "Brennan's conjecture and the Mandelbrot set", International Mathematics Research Notices 1998 (1998), no. 12, pp. 589–600, DOI: 10.1155/S1073792898000373, https://doi.org/10.1155/S1073792898000373
3. James E. Brennan, "The Integrability of the Derivative in Conformal Mapping", Journal of the London Mathematical Society (2) 18 (1978), pp. 261–272, DOI: 10.1112/jlms/s2-18.2.261, https://doi.org/10.1112/jlms/s2-18.2.261
4. Håkan Hedenmalm, Serguei Shimorin, "Weighted Bergman spaces and the integral means spectrum of conformal mappings", Duke Mathematical Journal 127 (2005), no. 2, pp. 341–393, DOI: 10.1215/S0012-7094-04-12725-3, https://doi.org/10.1215/S0012-7094-04-12725-3
5. Alexandru Aleman, Athanasios Kouroupis, "Brennan's conjecture holds for semigroups of holomorphic functions", arXiv preprint arXiv:2409.15074 (2024, revised 2025), https://arxiv.org/abs/2409.15074
6. Jianjun Jin, "Complex exponential integral means spectra of univalent functions and the Brennan conjecture", arXiv preprint arXiv:2512.09330 (2025, revised 2026), https://arxiv.org/abs/2512.09330
7. Neil Dobbs, Jacek Graczyk, Nicolae Mihalache, "Hausdorff Dimension of Julia Sets in the Logistic Family", Communications in Mathematical Physics 399 (2023), no. 2, pp. 673–716, DOI: 10.1007/s00220-022-04568-7, arXiv:2007.07661, https://arxiv.org/abs/2007.07661