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Math-2217 · mathematics · Analysis

Research Problems in Function Theory — Problem 6.109

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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 an analytic univalent $f$ in the unit disk and $\lambda>0$, set \[I_{-\lambda}(r,f')=\left(\frac1{2\pi}\int_0^{2\pi}|f'(re^{i\theta})|^{-\lambda}\,d\theta\right)^{1/\lambda}.\] Determine the maximal growth rate as $r\uparrow1$. In particular, for $\lambda=2$, must it be $O((1-r)^{-1/2})$, the rate of the Koebe function?

Formulation references: [S1], [S2].

Short English statement

For an analytic univalent $f$ in the unit disk and $\lambda>0$, set \[I_{-\lambda}(r,f')=\left(\frac1{2\pi}\int_0^{2\pi}|f'(re^{i\theta})|^{-\lambda}\,d\theta\right)^{1/\lambda}.\] Determine the maximal growth rate as $r\uparrow1$. In particular, for $\lambda=2$, must it be $O((1-r)^{-1/2})$, the rate of the Koebe function?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2217-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2217

Brennan integrability: The manuscript’s result bounds M_{-2} by C_epsilon(1-r)^(-1-epsilon); Hayman 6.109 asks I_{-2}=sqrt(M_{-2})=O((1-r)^(-1/2)) without any epsilon loss. The exponent is reached only up to loss and the full no-loss bound is not proved.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Research Problems in Function Theory — Problem 6.109

OpenAI's claim

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.
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

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.

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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