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Math-2216 · mathematics · Analysis

Research Problems in Function Theory — Problem 6.96

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a conformal map $f:\mathbb D\to\mathbb D$, let its integral-means exponent be \[\beta_f(p)=\limsup_{r\uparrow1}\frac{\log\int_0^{2\pi}|f'(re^{i\theta})|^p\,d\theta}{\log(1/(1-r))},\qquad B(p)=\sup_f\beta_f(p).\] Is the universal spectrum equal to $-p-1$ for $p\leq-2$, to $p^2/4$ for $-2\leq p\leq2$, and to $p-1$ for $p\geq2$?

Formulation references: [S1], [S2].

Short English statement

For a conformal map $f:\mathbb D\to\mathbb D$, let its integral-means exponent be \[\beta_f(p)=\limsup_{r\uparrow1}\frac{\log\int_0^{2\pi}|f'(re^{i\theta})|^p\,d\theta}{\log(1/(1-r))},\qquad B(p)=\sup_f\beta_f(p).\] Is the universal spectrum equal to $-p-1$ for $p\leq-2$, to $p^2/4$ for $-2\leq p\leq2$, and to $p-1$ for $p\geq2$?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2216-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2216

The proposed spectrum formula requires $B(-1)=1/4$. The inverse-first-power theorem asserts an upper growth exponent strictly below $1/4$, uniformly over normalized univalent maps, and hence also for bounded images. Rescaling a bounded map into the disk changes an integral mean by a constant factor and preserves its limiting exponent. Thus this result contradicts the spectrum identity in the precise disk-to-disk class recorded here. One failed parameter is enough to disprove the proposed universal identity; the theorem does not compute the exact remaining spectrum values. The Brennan paper provides a separate inverse-square result.

Research Problems in Function Theory — Problem 6.96

OpenAI's claim

Inverse first-power means: OpenAI announces a universal growth exponent below $1/4$ for circular means of the inverse derivative of normalized injective holomorphic disk maps; applied to bounded maps, this gives a strict inequality at the spectrum parameter $-1$, contrary to Kraetzer’s proposed value.

Brennan integrability: The announced integrability range for a conformal map onto the disk is $4/3<s<4$ for the area integral of the $s$th derivative power; the paper also identifies the inverse-square integral-means exponent for the schlicht class as one.

Mathematical statement of the claimed result

Mathematical claim 1.
Write $\mathcal S=\{f:\mathbb D\to\mathbb C:\ f\text{ holomorphic and injective},\ f(0)=0,\ f^\prime(0)=1\}$ and $M_p(f,r)=(2\pi)^{-1}\int_{-\pi}^{\pi}|f^\prime(re^{i\theta})|^p\,d\theta$. The claim is that some $0<\epsilon<1/4$ and finite $C$, independent of $f$ and $r$, satisfy \[M_{-1}(f,r)\le C(1-r)^{-1/4+\epsilon}\quad(f\in\mathcal S,\ 1/2\le r<1).\] In particular the bounded-image subclass has spectrum $B_b(-1)=\sup_f\limsup_{r\uparrow1}\log M_{-1}(f,r)/\log(1/(1-r))<1/4$. The theorem bounds this exponent; it does not determine its exact value.
Mathematical claim 2.
For every $\epsilon>0$ there is a finite $C_\epsilon$ such that all injective holomorphic $f:\mathbb D\to\mathbb C$ normalized by $f(0)=0$ and $f^\prime(0)=1$ satisfy \[\frac1{2\pi}\int_{-\pi}^{\pi}|f^\prime(re^{i\theta})|^{-2}\,d\theta\le C_\epsilon(1-r)^{-1-\epsilon}\quad(1/2\le r<1).\] The supremum over these maps of the limiting logarithmic growth exponent is exactly $1$. If $W\subset\mathbb C$ is simply connected, its spherical boundary has at least two points, and $\varphi:W\to\mathbb D$ is conformal and bijective, then \[\int_W|\varphi^\prime|^s\,dA<\infty\quad(4/3<s<4).\] Equivalently $\int_{\mathbb D}|f^\prime|^t\,dA<\infty$ for every injective holomorphic $f$ and $-2<t<2/3$. These are strict intervals, with no endpoint-integrability assertion. The sharp-endpoint corollary exhibits the Koebe map $k(z)=z/(1-z)^2$, for which $\int_{\mathbb D}|k^\prime|^{-2}\,dA=\int_{\mathbb D}|k^\prime|^{2/3}\,dA=\infty$, so both Brennan endpoints fail for its inverse.

Outline of the argument

Inverse first-power means: Affine rebasings of upper-half-plane maps define a positive averaging operator. Hypothetically excessive growth produces a scale-compatible weighted probability law and identities for logarithmic-derivative moments. Critical points of a target potential are paired geometrically; their signed Jacobians have an incompatible expected sign, and the equality case is separately eliminated.

Brennan integrability: The proof represents inverse-square derivative growth by a positive operator acting on normalized half-plane maps. An eigenmeasure is averaged over translations and scales to produce a weighted rebasing law. Moment identities from this law are compared with a pairing of critical points of target potentials, ruling out any growth exponent larger than one; change of variables then gives the stated derivative integrability.

Formal verification

Family 072 describes Lean coverage for the Brennan exponent interval, inverse-square means, and the strict inverse-first-power spectrum comparison.

Inverse first-power means: The manuscript links Brennan.lean (declaration OAI.Brennan.main_theorem), BrennanSharp.lean (declaration OAI.Brennan.Sharp.sharp_endpoints), StrictMeans.lean (declaration OAI.StrictInverseFirstPower.main).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

Brennan integrability: The manuscript links Brennan.lean (declaration OAI.Brennan.main_theorem), BrennanSharp.lean (declaration OAI.Brennan.Sharp.sharp_endpoints), StrictMeans.lean (declaration OAI.StrictInverseFirstPower.main).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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