Brennan's conjecture: integrability of the derivative of a conformal map and the universal integral means spectrum at exponent $-2$

Brennan's conjecture (1978) asserts that for every simply connected domain $\Omega\ne\mathbb{C}$ and every conformal map $\phi:\Omega\to\mathbb{D}$ onto the unit disk, the derivative belongs to $L^{s}(\Omega)$ for every $4/3<s<4$ ; equivalently, in the language of the integral means spectrum of the Riemann map $\varphi:\mathbb{D}\to\Omega$ , the universal spectrum satisfies $B(-2)\le 1$ . Kayumov (2006) proves the inequality $\beta_{f}(-2)\le 1$ only for bounded univalent functions whose $\log f'$ is a Hadamard-lacunary power series with lacunarity ratio $q\ge 15$ , via a Bessel-function inequality, and leaves open the removal of this structural hypothesis, the numerical effectiveness of the Jones–Makarov constant $A$ in $\beta_{f}(2-\varepsilon)\le 1-\varepsilon+A\varepsilon^{2}$ , and the existence of a universal constant $C$ uniform over the lacunary class. The conjecture remains open in general, with recent progress confined to special classes such as quasidisks with controlled Beltrami coefficient.

Background

A holomorphic function that is one-to-one on the unit disk $\mathbb{D}$ is called univalent; the class of normalized univalent functions ( $f(0)=0$ , $f'(0)=1$ ) is denoted $S$ . For such $f$ and a real exponent $t$ , the integral means spectrum is $\beta_{f}(t)=\limsup_{r\to1^{-}}\frac{\log\int_{-\pi}^{\pi}|f'(re^{i\theta})|^{t}\,d\theta}{-\log(1-r)}$ , and the universal integral means spectrum is $B(t)=\sup_{f}\beta_{f}(t)$ , the supremum over the class. These exponents measure the boundary growth of derivatives of conformal maps and control, for example, the dimension-type behavior of harmonic measure. In 1978 Brennan studied the integrability of the derivative of a conformal map $\phi:\Omega\to\mathbb{D}$ from a simply connected domain $\Omega\ne\mathbb{C}$ onto the unit disk and proved that $\phi'\in L^{s}(\Omega)$ for $4/3<s<3+\delta$ for some $\delta>0$ , extending the range $4/3<s<3$ that follows comparatively easily from the Koebe distortion theorem; he conjectured the sharp statement that $\phi'\in L^{s}(\Omega)$ for all $4/3<s<4$ . The slit domain $\Omega=\mathbb{C}\setminus(-\infty,-1/4]$ , the image of the Koebe function $k(z)=z/(1-z)^{2}$ (the standard extremal function of the class $S$ ), shows that this range cannot be extended. In the language of the integral means spectrum this is exactly $B(-2)\le 1$ : for the Riemann map $\varphi=\phi^{-1}:\mathbb{D}\to\Omega$ , the change of variables $w=\phi(z)$ gives $\int_{\Omega}|\phi'(z)|^{s}\,dA(z)=\int_{\mathbb{D}}|\varphi'(w)|^{2-s}\,dA(w)$ , so the integrability range $4/3<s<4$ becomes $-2<t<2/3$ for $t=2-s$ , and the critical endpoint $t=-2$ is the statement that the power-( $-2$ ) integral means grow no faster than $(1-r)^{-1-\varepsilon}$ for every $\varepsilon>0$ . Brennan also proved $B(t)<|t|$ for $t\le-2$ and conjectured the exact asymptotic $B(t)=|t|-1$ for $t\to-\infty$ . A power series $\sum a_{k}z^{n_{k}}$ is called Hadamard-lacunary with ratio $q>1$ if $n_{k+1}/n_{k}\ge q$ for all $k$ ; lacunary series behave in many respects like independent variables, which makes them tractable. Known partial results: Carleson and Makarov (1994) proved $B(t)=|t|-1$ for all sufficiently negative $t$ and recast the conjecture geometrically; Carleson and Jones (1992) proved $\beta_{f}(2)\le 1-\frac{3\pi}{4\log q}$ for Hadamard-lacunary $\log f'$ with ratio $q>2$ ; Jones and Makarov (1995) proved the near-endpoint bound $\beta_{f}(2-\varepsilon)\le 1-\varepsilon+A\varepsilon^{2}$ for bounded univalent $f$ , where the absolute constant $A$ is not bounded above by any concrete number; and Barański, Volberg and Zdunik (1998) established the conjecture for basins of infinity of quadratic polynomials, connecting it to the Mandelbrot set. Kayumov (2006) proves Brennan's conjecture, $\beta_{f}(-2)\le 1$ , under the structural hypothesis that $\log f'$ is Hadamard-lacunary with ratio $q\ge 15$ ; the proof rests on an inequality for the modified Bessel function $I_{0}$ (the power-series solution of $x^{2}y''+xy'-x^{2}y=0$ , which enters through Fourier estimates of lacunary exponentials) with constant $C=370$ valid for $0<x\le\frac{3\pi}{2}$ . Kayumov (2006) leaves open: the general conjecture beyond the lacunary class; the numerical effectiveness of the Jones–Makarov constant $A$ , on which any quantitative lowering of the threshold $15$ depends; the question whether a single constant $C$ can serve uniformly for the whole lacunary class in the bound $\int_{-\pi}^{\pi}|f'(re^{i\theta})|^{-2}\,d\theta\le C/(1-r)$ ; and the removal of auxiliary hypotheses (coefficient-size bounds, decay assumptions on the $a_{k}$ ).

Problem Statement

Prove or disprove Brennan's conjecture in its full generality: for every simply connected domain $\Omega\ne\mathbb{C}$ and every conformal map $\phi:\Omega\to\mathbb{D}$ onto the unit disk, the derivative satisfies $\phi'\in L^{s}(\Omega)$ , with respect to area measure on $\Omega$ , for every $4/3<s<4$ ; equivalently, for the Riemann map $\varphi=\phi^{-1}:\mathbb{D}\to\Omega$ in the language of the integral means spectrum, the universal spectrum obeys $B(-2)\le 1$ . Kayumov (2006) established this only under the structural hypothesis that $\log f'$ is Hadamard-lacunary with ratio $q\ge 15$ , so the essential target is the removal of all such structural hypotheses, i.e. the bound for arbitrary univalent functions (which would subsume non-lacunary series and any smaller lacunarity ratio; a bare lowering of the threshold $q\ge 15$ is a separate program).

References

I. R. Kayumov, Integral means spectrum and the modified Bessel function of zero order, St. Petersburg Mathematical Journal 17(3) (2006), 453–463, DOI 10.1090/s1061-0022-06-00914-9, https://doi.org/10.1090/s1061-0022-06-00914-9 (Russian original: Algebra i Analiz 17:3 (2005), 107–123)
James E. Brennan, The integrability of the derivative in conformal mapping, Journal of the London Mathematical Society (2) 18(2) (1978), 261–272, DOI 10.1112/jlms/s2-18.2.261, https://doi.org/10.1112/jlms/s2-18.2.261
Lennart Carleson and Nikolai G. Makarov, Some results connected with Brennan's conjecture, Arkiv för Matematik 32(1) (1994), 33–62, DOI 10.1007/BF02559522, https://doi.org/10.1007/BF02559522
Lennart Carleson and Peter W. Jones, On coefficient problems for univalent functions and conformal dimension, Duke Mathematical Journal 66(2) (1992), 169–206, DOI 10.1215/S0012-7094-92-06605-1, https://doi.org/10.1215/S0012-7094-92-06605-1
Peter W. Jones and Nikolai G. Makarov, Density properties of harmonic measure, Annals of Mathematics (2) 142(3) (1995), 427–459, DOI 10.2307/2118551, https://doi.org/10.2307/2118551
Krzysztof Barański, Alexander Volberg and Anna Zdunik, Brennan's conjecture and the Mandelbrot set, International Mathematics Research Notices 1998(12) (1998), 589–600, DOI 10.1155/S1073792898000373, https://doi.org/10.1155/S1073792898000373
Haakan Hedenmalm, Bloch functions and asymptotic tail variance, Advances in Mathematics 313 (2017), 947–990, DOI 10.1016/j.aim.2017.04.016, https://doi.org/10.1016/j.aim.2017.04.016
Jianjun Jin, On a multiplier operator induced by the Schwarzian derivative of univalent functions, Bulletin of the London Mathematical Society 56(7) (2024), 2296–2314, DOI 10.1112/blms.13056, https://doi.org/10.1112/blms.13056 (preprint: arXiv:2211.01742, https://arxiv.org/abs/2211.01742)
I. R. Kayumov, D. V. Maklakov and F. D. Kayumov, Estimates of a spectrum of the integral means for lacunary series, Russian Mathematics 58(10) (2014), 67–72, DOI 10.3103/S1066369X14100090, https://doi.org/10.3103/S1066369X14100090 (Russian original: Izvestiya VUZ. Matematika 2014(10), 79–85)
N. A. Shirokov, On means with power $-2$ for the dervatives [sic] of functions of class $S$ , St. Petersburg Mathematical Journal 28(6) (2017), 855–867, DOI 10.1090/spmj/1477, https://doi.org/10.1090/spmj/1477 (Russian original: Algebra i Analiz 28(6) (2016), 1189–1207)