---
title: Universal Integral Means Spectrum Definition
url: https://www.emergentmind.com/topics/universal-integral-means-spectrum
type: topic
---

# Universal Integral Means Spectrum Definition

The **universal integral means spectrum** is an extremal function describing the maximal boundary growth of derivative moments over normalized univalent functions. For the schlicht class
\[
\mathcal S=\{f:\mathbb D\to\mathbb C:\ f\text{ univalent},\ f(0)=0,\ f'(0)=1\},
\]
the integral means spectrum of \(f\) is
\[
\beta_f(t)=\limsup_{r\to1^-}
\frac{\log\displaystyle\int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta}
{|\log(1-r)|},
\qquad t\in\mathbb R,
\]
and the universal spectrum is
\[
B(t)=\sup_{f\in\mathcal S}\beta_f(t).
\]
The bounded universal spectrum is
\[
B_b(t)=\sup_{f\in\mathcal S_b}\beta_f(t),
\]
where \(\mathcal S_b\) is the subclass of bounded normalized univalent functions. These spectra encode extremal conformal distortion, harmonic-measure multifractality, and boundary singularity structure. Their precise determination is connected with the Koebe function, Brennan’s conjecture, quasiconformal Teichmüller theory, SLE, Gaussian multiplicative chaos, and thermodynamic formalism.

## 1. Definitions, normalizations, and basic structure

For a conformal map \(f\) on the unit disk, the integral means
\[
M_t(f,r)=\int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta
\]
measure the angularly averaged growth of \(f'\) on circles approaching the boundary. The exponent \(\beta_f(t)\) records the leading power-law behavior:
\[
M_t(f,r)\doteq (1-r)^{-\beta_f(t)}
\]
in the logarithmic sense when the corresponding limit exists. Positive \(t\) emphasizes large values of \(|f'|\), while negative \(t\) emphasizes small values.

The universal spectrum takes the supremum over a class of maps rather than describing a single conformal map. The distinction between \(B\) and \(B_b\) reflects the different boundary behavior of bounded and unbounded schlicht maps. Makarov’s identity gives
\[
B(t)=\max\{B_b(t),3t-1\}.
\]
Thus the unbounded contribution \(3t-1\) must be compared with the bounded spectrum. This relation is one of the principal mechanisms by which estimates for bounded univalent functions yield information for the full schlicht class [2405.07683].

The spectrum is convex in \(t\), as follows from Hölder’s inequality. Convexity constrains possible phase transitions and forces affine continuation in regions where the extremal boundary singularity has saturated the available distortion bound. The value at \(t=0\) is always
\[
\beta_f(0)=0,
\qquad B(0)=B_b(0)=0.
\]

Two conventions are common. The unrooted means use \(M_t(f,r)\) directly. The rooted means use
\[
\left(\frac1{2\pi}\int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta\right)^{1/t}.
\]
For finite \(t\), the corresponding exponents differ by a factor of \(t\). Precise comparisons therefore require fixing the normalization before interpreting formulas.

A related averaged spectrum arises for random conformal maps:
\[
\overline\beta(t)=\limsup_{r\to1^-}
\frac{\log\displaystyle\int_0^{2\pi}
\mathbb E\!\left[|\phi'(re^{i\theta})|^t\right]\,d\theta}
{|\log(1-r)|}.
\]
The expectation is taken before the logarithm and the limiting operation. This is an annealed spectrum and need not coincide with the almost-sure spectrum of a realization, as emphasized in the analysis of whole-plane SLE [1605.03112].

## 2. Universal extremals and the Koebe comparison

The Koebe function
\[
\kappa(z)=\frac{z}{(1-z)^2}
\]
is the fundamental extremal map for integral means in the normalized schlicht class. Baernstein’s star-function theorem states that, for \(f\in\mathcal S\),
\[
(\pm\log|f|)^*\le(\pm\log|\kappa|)^*,
\]
where the star-function compares the largest angular integrals over sets of prescribed measure. Consequently,
\[
M_t(f,r)\le M_t(\kappa,r)
\]
for the relevant positive and negative exponents. This is a pointwise-in-\(r\) comparison and is stronger than an asymptotic inequality. It implies that the Koebe function supplies the natural extremal comparison for universal integral means [1902.01595].

The explicit Koebe spectrum is
\[
\beta_\kappa(t)=
\begin{cases}
3t-1,&t\ge \frac13,\\[1mm]
0,&-1\le t<\frac13,\\[1mm]
|t|-1,&t<-1.
\end{cases}
\]
Indeed, \(|\kappa'(z)|\asymp |1-z|^{-3}\), and integration of powers of \(|1-re^{i\theta}|\) yields the displayed branches [2405.07683].

The Koebe comparison does not by itself determine \(B(t)\) throughout the entire parameter range. It identifies an extremal function for the integral means inequalities, while the universal spectrum also involves the distinction between bounded and unbounded maps and the possibility of sharper extremals in different classes. Known relations include
\[
B(t)=3t-1,\qquad t\ge\frac25,
\]
and
\[
B_b(t)=t-1,\qquad t\ge2.
\]
For \(0<t\le2/5\), one has
\[
B(t)>\frac{t^2}{5},
\]
while for sufficiently negative \(t\),
\[
B_b(t)=B(t)=|t|-1.
\]

The conjectural Kraetzer formula for the bounded spectrum is
\[
B_b(t)=
\begin{cases}
t^2/4,&|t|\le2,\\[1mm]
|t|-1,&|t|>2.
\end{cases}
\]
The quadratic branch is interpreted as a multifractal high-temperature regime, and the linear branches as frozen or terminating regimes. The status of this formula is not uniform across the parameter interval: the quadratic upper bound is established in the range \([-2,2]\), but equality in the interior has been challenged by recent work [2610.23853].

## 3. Convolution, star-functions, and comparison principles

For analytic functions
\[
f(z)=\sum_{n=0}^\infty a_nz^n,\qquad
g(z)=\sum_{n=0}^\infty b_nz^n,
\]
the Hadamard convolution is
\[
(f\star g)(z)=\sum_{n=0}^\infty a_nb_nz^n.
\]
A function \(F\) is bound preserving if
\[
f\star F\in H^\infty,\qquad
\|f\star F\|_{H^\infty}\le \|f\|_{H^\infty}
\]
for every \(f\in H^\infty\). Such functions admit a representation
\[
F(z)=\int_{\mathbb T}\frac{d\mu(\xi)}{1-z\xi},
\qquad \|\mu\|\le1,
\]
and satisfy
\[
(f\star F)(z)=\int_{\mathbb T}f(\xi z)\,d\mu(\xi).
\]

Minkowski’s inequality gives
\[
M_p(r,f\star F)\le M_p(r,f),
\qquad 1\le p\le\infty.
\]
Therefore convolution by a bound-preserving factor cannot increase the corresponding integral-means exponents in the range \(p\ge1\).

The Baernstein star-function is defined for a subharmonic function \(u\) by
\[
u^*(re^{i\theta})
=\sup_{|E|=2\theta}\int_Eu(re^{it})\,dt.
\]
Star-majorization is stronger than ordinary integral-means domination. If
\[
(\log|f|)^*\le(\log|g|)^*,
\]
then
\[
M_p(r,f)\le M_p(r,g)
\]
for every \(p>0\). However, star-function inequalities are not generally preserved under convolution. There exist analytic functions \(F_1,F_2\) individually dominated by the identity in the star order for which
\[
(\log|F_1\star F_2|)^*
\not\le
(\log|I\star I|)^*.
\]
There are also univalent convexity-preserving factors \(F\) for which
\[
(\log|f\star F|)^*\not\le(\log|f|)^*.
\]

A positive result holds for the class \(Z\) of Steiner symmetric univalent functions. If \(f\in Z\) and \(F\) is convexity preserving, then
\[
M_p(r,f\star F)\le M_p(r,f),
\qquad p>0.
\]
The additional geometric symmetry permits the extension of convolution contraction from the Banach range \(p\ge1\) to the quasi-Banach range \(0<p<1\) [1902.01595].

## 4. Teichmüller theory and nonattainment of bounded extremals

The universal Teichmüller space parametrizes normalized quasiconformal extensions of schlicht functions. If \(\mu\) is a Beltrami coefficient on \(\Delta^*\), let \(f_\mu\) be the normalized quasiconformal map that is conformal on \(\Delta\). The universal Teichmüller space is
\[
T=\mathcal M(\Delta^*)/\sim,
\]
where equivalent coefficients induce the same normalized function on \(\Delta\).

For each fixed \(t\), the integral means spectrum induces a functional
\[
I_T([\mu])=\beta_{f_\mu}(t).
\]
This functional is continuous in the Teichmüller metric. It is also continuous on the closure of the universal Teichmüller curve modeled by pre-Schwarzian derivatives
\[
N_f=\frac{f''}{f'}
\]
in the space
\[
E_1=\left\{\phi\in\mathcal A(\Delta):
\sup_{z\in\Delta}|\phi(z)|(1-|z|^2)<\infty\right\}.
\]
A pre-Schwarzian \(\phi\) reconstructs a normalized function through
\[
f_\phi(z)=
\int_0^z\exp\left(\int_0^\zeta\phi(w)\,dw\right)d\zeta.
\]

The asymptotic universal Teichmüller space identifies coefficients whose difference tends to zero at the boundary. The integral means spectrum is invariant under this asymptotic equivalence:
\[
\mu\sim_{\mathrm{as}}\nu
\quad\Longrightarrow\quad
\beta_{f_\mu}(t)=\beta_{f_\nu}(t).
\]
Consequently, the spectrum defines a continuous functional on the asymptotic Teichmüller space as well [2405.07683].

A principal strictness theorem states that every fixed quasiconformally extendible function lies strictly below the bounded universal spectrum:
\[
\beta_f(t)<B_b(t),
\qquad t\ne0,
\]
for \(f\in\mathcal S_q\), the class of bounded normalized schlicht functions admitting quasiconformal extensions. At the same time,
\[
B_b(t)=\sup_{f\in\mathcal S_q}\beta_f(t).
\]
Thus the supremum can be approached by quasiconformally extendible functions but is not attained by any individual member of that class.

The proof uses a multiplicative deformation. Given \(f\), one constructs a quasiconformally extendible map \(F\) satisfying
\[
F'(z)=[f'(z)]^{1+\varepsilon}.
\]
Then
\[
\beta_F(t)=\beta_f((1+\varepsilon)t),
\]
and strict monotonicity of the spectrum on the relevant positive or negative tail yields
\[
\beta_F(t)>\beta_f(t).
\]
This shows that any extremal, if it exists, must lie on the boundary of the relevant Teichmüller model rather than in its quasiconformal interior.

## 5. Brennan’s conjecture and the bounded universal spectrum

Brennan’s conjecture concerns the integrability of the derivative of a conformal map \(g:\Omega\to\mathbb D\):
\[
\int_\Omega |g'(w)|^p\,dA(w)<\infty,
\qquad 3<p<4.
\]
Writing \(f=g^{-1}\), a change of variables gives
\[
\int_\Omega |g'(w)|^p\,dA(w)
=
\int_{\mathbb D}|f'(z)|^{2-p}\,dA(z).
\]
The endpoint formulation in terms of integral means is
\[
B(-2)=1.
\]

A signed-area and critical-point argument establishes the bounded-map estimate
\[
B_b(t)\le \frac{t^2}{4},
\qquad -2\le t\le2.
\]
At \(t=-2\), this yields
\[
B_b(-2)\le1.
\]
Makarov’s identity then gives
\[
B(-2)=\max\{B_b(-2),-7\}\le1.
\]
Slit-type examples provide the opposite inequality, so
\[
B(-2)=1.
\]
This is the integral-means formulation of Brennan’s conjecture [2610.23853].

The proof introduces, for a bounded univalent map \(f\), the potential
\[
V_\xi(z)=(1-|z|^2)^k|f(z)-\xi|
\]
and the associated map
\[
G(z)=f(z)+\frac{(1-|z|^2)f'(z)}{2k\overline z}.
\]
Critical points of \(V_\xi\) correspond to solutions of \(G(z)=\xi\). The sign of the Jacobian \(\det DG\) distinguishes local maxima from saddles of the potential. An index theorem shows that maxima dominate saddles. Integrating this count yields a signed-area inequality, which is converted into a weighted radial estimate for
\[
M_t(f,r)=\int_0^{2\pi}|f'(re^{i\theta})|^t\,d\theta.
\]
Letting \(k\downarrow(2-t)/4\) produces the exponent \(t^2/4\).

The same work announces strictness,
\[
B_b(t)<\frac{t^2}{4},
\qquad t\in(-2,2)\setminus\{0\},
\]
which would disprove equality in Kraetzer’s conjecture throughout the nontrivial interior of the quadratic range. The non-strict bound is proved, whereas the detailed strict argument is stated to be deferred [2610.23853]. Earlier Teichmüller-theoretic work also proves strict inequality for each fixed quasiconformally extendible function, but that pointwise result does not by itself determine the universal envelope.

## 6. Model spectra: SLE, logarithmic tracts, and random fields

The universal spectrum is connected to several explicit but generally non-universal model spectra.

For bounded/exterior whole-plane \(\mathrm{SLE}_\kappa\), the averaged spectrum is
\[
\overline\beta(t)=
\begin{cases}
\beta_{\mathrm{tip}}(t),&t\le t_2,\\
\beta_0(t),&t_2\le t\le t_3,\\
\beta_{\mathrm{lin}}(t),&t_3\le t,
\end{cases}
\]
where
\[
\beta_{\mathrm{tip}}(t)
=
-t-1+\frac14\left(4+\kappa-
\sqrt{(4+\kappa)^2-8\kappa t}\right),
\]
\[
\beta_0(t)
=
-t+\frac{4+\kappa}{4\kappa}
\left(4+\kappa-\sqrt{(4+\kappa)^2-8\kappa t}\right),
\]
and
\[
\beta_{\mathrm{lin}}(t)
=
t-\frac{(4+\kappa)^2}{16\kappa}.
\]
The transition parameters are
\[
t_2=-1-\frac{3\kappa}{8},
\qquad
t_3=\frac{3(4+\kappa)^2}{32\kappa}.
\]
The tip branch dominates for sufficiently negative moments, the bulk branch describes generic boundary behavior, and the linear branch results from convexity and freezing. In the bounded formulation, a second-point spectrum associated with the origin is mathematically present but is hidden in the full angular integral by the earlier tip transition [1605.03112].

For interior whole-plane SLE, a generalized spectrum incorporates the image-size parameter \(q\):
\[
\beta(p,q)
=
\limsup_{r\to1^-}
\frac{\log\displaystyle\int_{|z|=r}
\mathbb E\!\left(
|z|^q\frac{|f'(z)|^p}{|f(z)|^q}
\right)|dz|}
{-\log(1-r)}.
\]
The line \(q=0\) gives the usual interior spectrum, while \(q=2p\) corresponds to the bounded exterior formulation. The conjectured phase diagram contains tip, bulk, infinity, and linear branches. The infinity branch is
\[
\beta_1(p,q)
=
3p-2q-\frac12-\frac12\sqrt{1+2\kappa(p-q)}.
\]
Rigorous equality with this branch has been extended to a region \(\mathfrak D_1\cap\mathcal I\), and for \(q=0\) to
\[
p\in[p^*(\kappa),p(\kappa)].
\]
The full four-phase formula remains conjectural [2203.10782].

A different tract-dependent spectrum is defined for logarithmic tracts of entire functions in the Eremenko–Lyubich class \(\mathcal B\). For rescaled tract maps \(\varphi_T\),
\[
\beta_\infty(t)
=
\limsup_{T\to\infty}
\beta_{\varphi_T}(1/T,t),
\qquad
b_\infty(t)=\beta_\infty(t)-t+1.
\]
This is not the universal schlicht spectrum. It measures the fractal geometry of a fixed tract at infinity and controls transfer-operator convergence. The critical value
\[
\Theta=\inf\{t>0:b_\infty(t)=0\}
\]
separates convergence and divergence of the transfer operator. Hölder tracts have negative spectrum, and the resulting thermodynamic formalism yields pressure, conformal measures, Gibbs states, exponential mixing, a central limit theorem, and Bowen’s formula for the hyperbolic dimension [1709.05166].

Random analytic models can reproduce the quadratic-to-linear profile associated with Kraetzer’s conjecture without belonging to the univalent class. The primitive of the randomized Riemann zeta function and the primitive of a holomorphic Gaussian multiplicative chaos derivative have almost surely
\[
\beta(\tau)=
\begin{cases}
|\tau|^2/4,&|\tau|\le2,\\[1mm]
|\tau|-1,&|\tau|\ge2.
\end{cases}
\]
The quadratic branch arises from log-correlated fluctuations, while the linear branch is a freezing transition. Both primitives are almost surely non-injective, so these results are analogues of the universal spectrum rather than proofs about normalized univalent functions [2603.26507].

The universal integral means spectrum therefore occupies a boundary position between extremal conformal geometry and a broad collection of model-specific theories. Its rigorous content includes sharp comparison principles, Teichmüller continuity, nonattainment within quasiconformal classes, the proof of Brennan’s conjecture, and extensive exact spectra for SLE, logarithmic tracts, Lévy–Loewner evolutions, and log-correlated random fields. Its complete determination, including the status of the interior quadratic regime and the existence or structure of universal extremals, remains a central problem in geometric function theory.

Source: https://www.emergentmind.com/topics/universal-integral-means-spectrum