Exact values and extremal functions of the universal integral means spectrum for complex exponents (the Kraetzer, Binder, and Brennan conjectures)

For a univalent function $f$ on the unit disk, the integral means spectrum $\beta_f(\tau)$ measures the boundary growth of $\int|f'|^{\tau}$; for complex exponent $\tau$ it also encodes rotation. The universal spectra $B(\tau)=\sup_{f\in\mathcal S}\beta_f(\tau)$ and $B_b(\tau)=\sup_{f\in\mathcal S_b}\beta_f(\tau)$ over the normalized and bounded univalent classes are among the central quantitative invariants of conformal mapping, yet their exact values are unknown: Brennan's conjecture (1978) asserts $B(-2)=B_b(-2)=1$, Kraetzer's conjecture predicts $B_b(\tau)=\tau^2/4$ on $|\tau|\le2$ and $|\tau|-1$ outside for real $\tau$, and Binder's conjecture extends the same piecewise formula, read with $|\tau|$ in place of the real variable, to complex $\tau$. A 2025-2026 preprint of Jianjun Jin determined the integral means spectrum, for real exponents, of every univalent rational function (so Brennan's conjecture holds for that class and for polygon maps), proved that no function with a quasiconformal extension can be extremal, and posed the complementary global problem: for each $\tau\ne0$, does an extremal function for $B_b(\tau)$ exist, and in which part of $\mathcal S$ does it live? This record formulates the combined open problem: determine $B_b(\tau)$ exactly for complex $\tau$ and settle the existence and Teichmuller-space location of its extremals. Both the value conjectures and the extremal-existence question remain open in general.

Background

A holomorphic function on the unit disk $\Delta=\{z:|z|<1\}$ is called univalent if it is one-to-one, i.e. a conformal mapping onto its image. Write $\mathcal S$ for the class of univalent $f$ with $f(0)=0$ and $f'(0)=1$ (normalization), and $\mathcal S_b\subset\mathcal S$ for the bounded ones. For a real or complex exponent $\tau$ one defines $[f'(z)]^{\tau}=\exp(\tau\log f'(z))$, which is single-valued since $f'\neq0$ in $\Delta$ and $f'(0)=1$; complex $\tau$ tracks rotation of the image boundary in addition to expansion and compression. The integral means spectrum of $f$ is $\beta_f(\tau)=\limsup_{r\to1^-}\frac{\log\int_{-\pi}^{\pi}\bigl|[f'(re^{i\theta})]^{\tau}\bigr|\,d\theta}{|\log(1-r)|},$ and the universal integral means spectra are $B(\tau)=\sup_{f\in\mathcal S}\beta_f(\tau)$ and $B_b(\tau)=\sup_{f\in\mathcal S_b}\beta_f(\tau)$. Because the integral means of $f'$ control harmonic measure in the image domain, these functionals are central to the fine boundary theory of conformal maps.

For real $\tau$ the known landmarks are as follows. Makarov proved the reduction $B(\tau)=\max\{B_b(\tau),\,3\tau-1\}$; Feng and MacGregor established the linear law $B(\tau)=3\tau-1$ for all sufficiently large positive $\tau$, and Pommerenke proved $B_b(\tau)=\tau-1$ for $\tau\ge2$. On the negative side, Carleson and Makarov showed that there is a threshold $\tau^*<0$ with $B(\tau)=B_b(\tau)=|\tau|-1$ for all $\tau\le\tau^*$, and only $\tau^*\le-2$ is known. The conjecture that in fact $\tau^*=-2$ is equivalent to Brennan's conjecture (1978) that $B(-2)=B_b(-2)=1$, one of the most studied open problems of geometric function theory; the sharpest known upper bound for $B(-2)$ is due to Hedenmalm and Shimorin, obtained through weighted Bergman space techniques. Carleson and Jones conjectured $B_b(1)=1/4$, and Kraetzer, from numerical experiments, proposed the unifying piecewise formula $B_b(\tau)=\tau^2/4$ for $|\tau|\le2$ and $B_b(\tau)=|\tau|-1$ outside. For complex exponents much less is known: Becker and Pommerenke asked whether $B_b(\tau)=1$ whenever $|\tau|=2$ (the complex form of Brennan's conjecture); Binder extended Makarov's reduction to complex $\tau$ and proved that along each ray $\tau=te^{i\theta}$ one has $B_b(\tau)=t-1$ for all sufficiently large $t$; Baranov and Hedenmalm obtained the nearly sharp second-order bound $B_b(2-\tau)\le1-\mathrm{Re}(\tau)+(\tfrac{9e^2}{2}+o(1))|\tau|^2\log\tfrac{1}{|\tau|}$ as $|\tau|\to0$, i.e. near the point $\tau=2$. Binder also conjectured that Kraetzer's formula, read with $|\tau|$ in place of the real variable, holds for every complex $\tau$.

By the fractal approximation principle (the principle of Makarov and of Carleson and Jones that, for the universal spectra, arbitrary image domains may be replaced by domains bounded by quasicircles without changing the supremum), $B_b(\tau)$ is already the supremum of $\beta_f(\tau)$ over the subclass $\mathcal S_q\subset\mathcal S_b$ of bounded univalent functions that extend to a quasiconformal homeomorphism of the extended plane (a map whose Beltrami coefficient $\mu=\bar\partial f/\partial f$ satisfies $\|\mu\|_\infty<1$). This turns $B_b(\tau)$ into a global extremal problem on quasiconformal Teichmuller spaces: the universal Teichmuller space $T$ (equivalence classes of Beltrami coefficients on the exterior of $\Delta$, two coefficients being equivalent when they give the same map on the boundary circle), the universal asymptotic Teichmuller space $AT$ (coefficients identified when they agree asymptotically at the boundary), and the universal Teichmuller curve, modeled as the set $\mathcal T$ of Pre-Schwarzian derivatives. Here the Pre-Schwarzian derivative of a locally univalent $f$ is $N_f=f''/f'$ and the Schwarzian derivative is $S_f=(f'''/f')-\tfrac32(f''/f')^2$; both are classical rigidity invariants of conformal maps. In a series of works (2024-2026), Jianjun Jin proved that the integral means spectrum functional $f\mapsto\beta_f(\tau)$ is continuous on $T$, $AT$, and the closure $\overline{\mathcal T}$ for every complex $\tau$, and proved the strict inequality $\beta_f(\tau)<B_b(\tau)$ for every $f\in\mathcal S_q$ and $\tau\ne0$: no quasiconformally extendable function can be an extremal for $B_b(\tau)$. Known extremals exist only in the outer ranges: $f(z)=-\log(1-z)$ is extremal for $B_b(\tau)$ when $\tau\ge2$, and $f(z)=z-\tfrac12z^2$ when $\tau\le\tau^*$; in the critical window $0<|\tau|<2$ no candidate is known. In the same preprint, Jin completely determined the integral means spectra, for real exponents, of all univalent rational functions (in particular, the Brennan conjecture holds for that class, and for univalent maps onto the interior of a bounded polygon), and verified the multiplier criterion of Shimorin for further classes: Shimorin showed that if the norm $M_f(\alpha)$ of the multiplication operator $\varphi\mapsto S_f\varphi$ from the weighted Bergman space $H^2_\alpha(\Delta)$ (the holomorphic functions with $\int|\varphi|^2(1-|z|^2)^\alpha\,dA<\infty$) to $H^2_{\alpha+4}(\Delta)$ satisfies the Koebe-type bound $M_f(\alpha)^2\le36(\alpha+3)(\alpha+5)/((\alpha+2)(\alpha+4))$ for all $\alpha>0$, then $\beta_f(-2)\le1$, i.e. Brennan's conjecture holds for that $f$; the Koebe function $\kappa(z)=z/(1-z)^2$ and its rotations saturate the bound. Jin proved that the specific rational maps $z-\tfrac12z^2$ and $z-\tfrac13z^3$ also saturate it, and proved in a companion paper the reverse inequality $M_R(\alpha)\ge M_\kappa(\alpha)$ for every univalent rational function $R$ with a critical point on the unit circle; whether $M_f(\alpha)\le M_\kappa(\alpha)$ holds for all $f\in\mathcal S$ (which would imply Brennan's conjecture) remains open. Jin posed the problems and conjectures assembled in this record.

Problem Statement

Let $\mathcal S$ be the class of univalent functions $f$ on the unit disk with $f(0)=0$, $f'(0)=1$, let $\mathcal S_b\subset\mathcal S$ be the bounded ones, and for complex $\tau$ let $\beta_f(\tau)=\limsup_{r\to1^-}\frac{\log\int_{-\pi}^{\pi}\bigl|[f'(re^{i\theta})]^{\tau}\bigr|\,d\theta}{|\log(1-r)|},\qquad B_b(\tau)=\sup_{f\in\mathcal S_b}\beta_f(\tau),\qquad B(\tau)=\sup_{f\in\mathcal S}\beta_f(\tau).$ These two components form a single global extremal problem for the integral means spectrum functional, as posed in the source paper: the value of the extremal problem, and the existence and location of its optimizer. The two sub-parts below are posed jointly: (a) determines the values $B_b(\tau)$ for all $\tau\in\mathbb C$, while (b) asks, for each $\tau$ for which it is not already settled, whether that value is attained and, if so, in which subclass the extremal lies. Of the location claims in (b), Conjecture 7.11 follows from Conjecture 7.10, so the open decisions there are Conjectures 7.10 and 7.25.

(a) Values. Determine the exact value of $B_b(\tau)$ for every $\tau\in\mathbb C$. In particular, decide Binder's conjecture (the complex extension of Kraetzer's conjecture) that $B_b(\tau)=|\tau|^2/4$ for $|\tau|<2$ and $B_b(\tau)=|\tau|-1$ for $|\tau|\ge2$; the source paper states the first branch as $\tau^2/4$, which for complex $\tau$ is to be read with the modulus, consistent with the real case, the $|\tau|-1$ branch, and the literature form $B_k(t)=k^2|t|^2/4$ of the $k$-quasiconformal analogue. Restricted to real $\tau$ this is Kraetzer's conjecture, its point $\tau=-2$ is Brennan's conjecture $B_b(-2)=B(-2)=1$, and its circle $|\tau|=2$ is the complex Brennan question of Becker and Pommerenke.

(b) Extremals. Call $f\in\mathcal S$ an extremal function for $B_b(\tau)$ if $\beta_f(\tau)=B_b(\tau)$. For each fixed $\tau\ne0$: does an extremal function for $B_b(\tau)$ exist, and if so, in which subclass of $\mathcal S$ does it lie? In particular, decide whether, for each $\tau\in\mathbb C$, there is an extremal whose Pre-Schwarzian derivative $N_f=f''/f'$ lies on the boundary of the universal Teichmuller curve $\mathcal T$ (the space of Pre-Schwarzian derivatives of the functions in $\mathcal S$; the source paper calls such an extremal regular, Conjecture 7.10), which would imply that there is one whose Schwarzian derivative $S_f=(f'''/f')-\tfrac32(f''/f')^2$ lies on the boundary of the Schwarzian image $S_Q$ of the quasiconformally extendable univalent class (Conjecture 7.11), and decide whether there is one whose Pre-Schwarzian class lies on the boundary of the universal asymptotic Teichmuller space (Conjecture 7.25). Known: extremals cannot lie in the class $\mathcal S_q$ of bounded univalent functions admitting a quasiconformal extension to the extended plane (Jin's strict inequality $\beta_f(\tau)<B_b(\tau)$ for every such $f$ and $\tau\ne0$), and the outer ranges $\tau\ge2$ and $\tau\le\tau^*$ do admit extremals, where $\tau^*$ is the negative threshold of Carleson and Makarov below which $B_b(\tau)=|\tau|-1$ is proved (only $\tau^*\le-2$ is known, conjecturally $\tau^*=-2$); the question is open for the remaining $\tau$, including the whole critical window $0<|\tau|<2$.

References

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