Brennan's conjecture on the $L^p$-integrability of the derivative of a conformal map and the simple-fraction expansion route

Brennan's conjecture (1978) asserts that if $\varphi$ maps a simply connected domain $\Omega$ whose boundary contains more than one point conformally onto the unit disk, then $\varphi'$ belongs to $L^p(\Omega)$ for every $4/3<p<4$; equivalently, for every conformal $f$ of the unit disk, $|f'|^p$ is area-integrable for every $-2<p<2/3$. The upper endpoint is sharp (Koebe function), and the best general result (Hedenmalm-Shimorin, 2005) reaches only $p>-1.752$, so the conjecture remains one of the central open problems of geometric function theory. The source paper (F. D. Kayumov, 2014) proves the conjecture under each of two sufficient conditions: a condition on the even Taylor coefficients of $\log f'$, and the existence of a simple-fraction expansion $1/f'(z)=\sum \alpha_k/(z-z_k)$ with $\sum|\alpha_k/z_k|<\infty$. Whether the simple-fraction condition holds for every univalent function is open, and an affirmative answer would prove the full conjecture; the even-coefficient condition, by contrast, is not satisfied by every univalent function (it already fails for the Koebe function), so it is not itself a route to the conjecture.

Background

A function holomorphic and one-to-one in the unit disk $\mathbb{D}$ is called univalent; by the Riemann mapping theorem, every simply connected domain $\Omega\subsetneq\mathbb{C}$ whose boundary contains more than one point is the image of $\mathbb{D}$ under a conformal (univalent) map. The class $S$ consists of the univalent functions $f(z)=z+c_2z^2+\cdots$ normalized by $f(0)=0$, $f'(0)=1$. The Koebe function $k(z)=z/(1-z)^2$, which maps $\mathbb{D}$ onto the plane cut along a half-line, is extremal for most growth questions; the classical distortion theorem for $S$ implies in particular $|1/f'(z)|\le(1+|z|)^3/(1-|z|)$.

Brennan (1978) conjectured that if $\varphi:\Omega\to\mathbb{D}$ is conformal, then $\varphi'\in L^p(\Omega)$ for every $4/3<p<4$, that is, $\iint_\Omega|\varphi'|^p\,dA<\infty$, where $dA$ denotes planar area measure. By the change of variables $w=f(z)$ this is equivalent to the disk formulation: for every conformal $f:\mathbb{D}\to\Omega$, $\iint_{\mathbb{D}}|f'(z)|^p\,dA<\infty\quad\text{for all }-2<p<2/3.$ The positive endpoint $2/3$ is sharp, as shown by the Koebe function, so the conjectural content is the negative endpoint $-2$ (equivalently, the endpoint $4$ for the inverse map). Pommerenke (1985) proved the conjecture equivalent to an integral-means estimate, where the integral means of the derivative are the circle averages $\int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta$: specifically, Brennan's conjecture holds if and only if $\int_0^{2\pi}|f'(re^{i\theta})|^{-2}\,d\theta=O((1-r)^{-1-\varepsilon})$ for every $\varepsilon>0$. Carleson and Makarov (1994) and Bertilsson (1999, who also proved the conjecture equivalent to the sharper bound $C|f'(0)|^{-2}/(1-r)$) obtained deep partial results, and a reduction recalled by Aleman and Kouroupis (2024), due to Makarov, shows it would suffice to verify the conjecture for bounded slit maps. The best known general range is due to Hedenmalm and Shimorin (2005): $|f'|^p$ is integrable for every $p>-1.752$. The conjecture is also known only for special classes, e.g. functions with nonnegative Taylor coefficients of $\log(zf'/f)$ (I. R. Kayumov, 2005), holomorphic semigroups, that is, one-parameter continuous semigroups of holomorphic self-maps of the disk (Aleman-Kouroupis, 2024), rational univalent functions (Jin, 2025), and basins of attraction at infinity (escaping sets) of certain polynomials (Zheng, 2026).

The source paper (F. D. Kayumov, 2014) approaches the conjecture through two sufficient conditions. Writing $\log f'(z)=\sum_{k\ge0}a_kz^k$, its Theorem 1 shows that Brennan's conjecture holds for $f$ if the even coefficients satisfy $\sum_{k\ge0}|a_{2k}|r^{2k}=o\bigl(\log\tfrac{1}{1-r}\bigr)$ as $r\to1$; this condition is sufficient but not universal, since for the Koebe function $k(z)=z/(1-z)^2$, whose $\log k'(z)=\log(1+z)-3\log(1-z)$ has even coefficients $a_{2k}=1/k$, one gets $\sum_{k\ge1}|a_{2k}|r^{2k}=\log\tfrac{1}{1-r}+O(1)$, which is not $o\bigl(\log\tfrac{1}{1-r}\bigr)$. A simple fraction is a rational function of the form $\alpha/(z-z_k)$ with a single pole; by a classical theorem of Wolff (1921), strengthened by Denjoy (1924), every function analytic in $\mathbb{D}$ admits an expansion as a series of simple fractions with poles outside the disk. Since $f'$ has no zeros in $\mathbb{D}$, $1/f'$ is analytic and hence admits such expansions; Theorem 2 of the source paper shows that if $1/f'(z)=\sum_{k\ge1}\alpha_k/(z-z_k)$ with $|z_k|>1$ and $\sum_{k\ge1}|\alpha_k/z_k|<\infty$, then $\int_0^{2\pi}|f'(re^{i\theta})|^{-2}d\theta\le C/(1-r^2)$, which is more than enough for Brennan's conjecture. The same paper proves that $1/f'$ can be approximated by $n$-term simple fractions with error $O(1/n)$ on compact subdisks, so approximability is not the obstruction; the open question is whether the expansion can always be chosen with summable coefficients.

Problem Statement

Settle Brennan's conjecture in its standard general form: is it true that for every conformal map $f$ of the unit disk $\mathbb{D}$ onto a simply connected domain whose boundary contains more than one point, $\iint_{\mathbb{D}}|f'(z)|^p\,dA(z)<\infty\quad\text{for every }p\text{ with }-2<p<2/3,$ equivalently, that for the inverse map $\varphi$ of the domain onto the disk, $\varphi'\in L^p(\Omega)$ for every $4/3<p<4$? The objective is to determine the exact integrability range of the derivative of a general conformal map. The source paper's other sufficient condition (Theorem 1) is not universal: the even-coefficient condition $\sum_{k\ge0}|a_{2k}|r^{2k}=o\bigl(\log\tfrac{1}{1-r}\bigr)$, $r\to1$, where $\log f'(z)=\sum_{k\ge0}a_kz^k$, fails for the Koebe function, so it does not hold for every univalent function.

References

1. F. D. Kayumov, "Integral estimates for derivatives of univalent functions", Lobachevskii Journal of Mathematics 35(4), 402-408 (2014). DOI: 10.1134/S1995080214040052, https://doi.org/10.1134/S1995080214040052. Russian original: Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki 155(2), 83-90 (2013), https://www.mathnet.ru/eng/uzku1208
2. J. E. Brennan, "The integrability of the derivative in conformal mapping", Journal of the London Mathematical Society (2) 18(2), 261-272 (1978). DOI: 10.1112/jlms/s2-18.2.261, https://doi.org/10.1112/jlms/s2-18.2.261
3. Ch. Pommerenke, "On the integral means of the derivative of a univalent function", Journal of the London Mathematical Society (2) 32(2), 254-258 (1985). DOI: 10.1112/jlms/s2-32.2.254, https://doi.org/10.1112/jlms/s2-32.2.254
4. L. Carleson and N. G. Makarov, "Some results connected with Brennan's conjecture", Arkiv för Matematik 32(1), 33-62 (1994). DOI: 10.1007/BF02559522, https://doi.org/10.1007/BF02559522
5. D. Bertilsson, "On Brennan's conjecture in conformal mapping", Doctoral Thesis, Royal Institute of Technology (KTH), Stockholm (1999). zbMATH: Zbl 0949.30013, https://zbmath.org/?q=an%3A0949.30013; full text: http://www.diva-portal.org/smash/get/diva2:8593/FULLTEXT01.pdf (DiVA record diva2:8593)
6. I. R. Kayumov, "On Brennan's conjecture for a special class of functions", Mathematical Notes 78(3-4), 498-502 (2005). DOI: 10.1007/s11006-005-0149-1, https://doi.org/10.1007/s11006-005-0149-1 (Russian original: Matematicheskie Zametki 78(4), 537-541; Springer/Crossref metadata for the English translation lists "I. P. Kayumov")
7. H. Hedenmalm and S. Shimorin, "Weighted Bergman spaces and the integral means spectrum of conformal mappings", Duke Mathematical Journal 127(2), 341-393 (2005). DOI: 10.1215/s0012-7094-04-12725-3, https://doi.org/10.1215/s0012-7094-04-12725-3
8. A. Aleman and A. Kouroupis, "Brennan's conjecture holds for semigroups of holomorphic functions", arXiv:2409.15074 (2024). https://arxiv.org/abs/2409.15074
9. J. Jin, "Complex exponential integral means spectra of univalent functions and the Brennan conjecture", arXiv:2512.09330 (2025). https://arxiv.org/abs/2512.09330
10. Y. Zheng, "Brennan Conjecture for Basin of Attraction at Infinity", arXiv:2604.12240 (2026). https://arxiv.org/abs/2604.12240
11. J. Wolff, "Sur les séries $\sum A_k/(z-a_k)$", Comptes Rendus de l'Académie des Sciences, Paris 173, 1056-1057 and 1327-1328 (1921). JFM identifier: 48.0320.01, https://zbmath.org/?q=an%3A48.0320.01; open scan: https://gallica.bnf.fr/ark:/12148/bpt6k31267.f1056
12. A. Denjoy, "Sur les séries de fractions rationnelles", Bulletin de la Société Mathématique de France 52, 418-434 (1924). DOI: 10.24033/bsmf.1061, https://doi.org/10.24033/bsmf.1061
13. I. R. Kayumov, "A survey on the integral means spectrum for conformal mappings" (in Russian), Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki 157(2), 104-115 (2015). MathNet: https://www.mathnet.ru/eng/uzku1310; zbMATH: Zbl 1342.30002, https://zbmath.org/?q=an%3A1342.30002