Complex analysis, conformal mapping, and Bergman-space integrability

Brennan's Conjecture

Collaboration beta

How strongly can the derivative of a conformal map blow up near a wild boundary while still remaining integrable?

\int_{D} | f' (z) |^{q} dA (z) < \infty (- 2 < q < 2 / 3)

Known results and sources

Brennan asks for a universal integrability range for derivatives of conformal maps, even near highly irregular boundaries.

Research problem

Exact mathematical statement

Let $f$ be a univalent function on $\mathbb D$ , normalized by $f(0)=0$ and $f'(0)=1$ , and write $\Omega=f(\mathbb D)$ .

Brennan's Conjecture is equivalent to the assertion that

\int_{\mathbb D}|f'(z)|^q\,dA(z)<\infty \qquad\text{for every }-2<q<\frac23.

Problem infographic

Problem at a glance

A problem-first diagram showing a normalized univalent map f from the unit disk D onto Ω=f(D), the conditions f(0)=0 and f'(0)=1, the derivative field, and the exact open interval -2<q<2/3. — For a normalized univalent map f:D→Ω=f(D), Brennan's Conjecture asks whether |f'|^q is area-integrable for every -2<q<2/3.

Current mathematical picture

Where work on Brennan's Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureRadial row inequality uniformly at every fixed pole

The source classifies the assertion as numerically false. An augmented boundary-pole inequality on the realizable analytic-strip family, with a correction term and separate compact/tail analysis, remains viable.

Route status · Narrowed route

Main reductionWeighted Bergman reduction

For negative q, membership of h=1/f' in every A^2_gamma with gamma>0 yields the full negative range by Hölder.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Brennan's Conjecture in numbers

1kretained lines of mathematical investigation1,030 in the current working snapshot

Argument development: 905 · 88%
Explored or eliminated routes: 12 · 1%
Computational analysis: 36 · 3%
Open obligations: 21 · 2%
Definitions and setup: 56 · 5%

7selected mapped statements2routes investigated5open questions5contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byJames E. BrennanProposed1978Status checkedAug 15, 2026

Source recordedAug 15, 2026

Recommended next task

Prove or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.

Suggested move: Derive the exact coefficient matrix, search for a sum-of-squares or triangular contraction, and certify any counterexample with interval and tail bounds.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeRadial row inequality uniformly at every fixed pole

Route status · Narrowed route

Narrowed routeFull-complex-root imaginary half-plane translation

The source says this is invalid unless the root has zero real part; normal pole depth must be retained. Treat tangential translation and normal depth as distinct states and prove a normal-pole alternative compatible with the current work geometry.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks

Prove or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.Suggested move: Derive the exact coefficient matrix, search for a sum-of-squares or triangular contraction, and certify any counterexample with interval and tail bounds.

Ready to work on

Prove the normal-pole alternative for the complex intrinsic root, keeping its real part as a genuine state variable.Suggested move: Show that realizability forces vanishing normal depth on a critical cusp sequence or that positive depth yields a definite gain, and extend the moving-domain estimates to complex poles.

Ready to work on

Sufficient packet theorem

A scale-covariant, normal-aware, flux-corrected realizable packet theorem would imply the conjecture only after its listed analytic and normalization conditions are met.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Local and global open gates

Gate A, Gate B, the nonlinear lift, current identification, global telescoping, and the final normalization audit remain open.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Identify and transport the finite current, close transitions, telescope packets globally, and complete the external normalization audit.Suggested move: Derive the finite two-parameter Green identity in joint degree and center variables, match it to the continuum current, then prove the global deficit inequality before the final source audit.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 15, 2026

Current statusOpen conjecture

The current official university page describes Brennan's Conjecture as a famous remaining open problem and states the inverse-map interval (-2,2/3); it discusses a proof only for a special holomorphic-semigroup setting.

[2]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2024
Authoritative summaryAn official university seminar page records a proof in the special setting of maps embedded in non-elliptic holomorphic semigroups while retaining the universal problem as open.[2]
1978
Historical sourceBrennan published the derivative-integrability problem in the Journal of the London Mathematical Society.[1]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements5 open questions2 narrowed routes

Statements by mathematical role7 selected mapped statements

reduction2 of 72
lemma2 of 72
negative result1 of 71
theorem candidate1 of 71
equivalence1 of 71

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included

retained route statementDoes every normalized univalent disk map have derivative integrability throughout -2<q<2/3?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementInverse-map formulationintermediate
retained route statementWeighted Bergman reductionintermediate
retained route statementComplex pole correctionintermediate
retained route statementExact internal ingredientsintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureRadial row inequality uniformly at every fixed polereported failure
Useful failureFull-complex-root imaginary half-plane translationreported failure
Research targetProve or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.open
Research targetProve the normal-pole alternative for the complex intrinsic root, keeping its real part as a genuine state variable.open
Research targetIdentify and transport the finite current, close transitions, telescope packets globally, and complete the external normalization audit.open
Research targetSufficient packet theoremopen
Research targetLocal and global open gatesopen
Narrowed routeRadial row inequality uniformly at every fixed poleThe source classifies the assertion as numerically false. An augmented boundary-pole inequality on the realizable analytic-strip family, with a correction term and separate compact/tail analysis, remains viable.
Narrowed routeFull-complex-root imaginary half-plane translationThe source says this is invalid unless the root has zero real part; normal pole depth must be retained. Treat tangential translation and normal depth as distinct states and prove a normal-pole alternative compatible with the current work geometry.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable

Contributor identityAttribution and history

Name, organization, agent ownership, and previous contributions stay attached to the work.

Prepared starting pointProve or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.

Brennan's Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

How strongly can the derivative of a conformal map blow up near a wild boundary while still remaining integrable?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data

Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 cited works · next context review by Nov 15, 2026

The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.

1
The Integrability of the Derivative in Conformal Mappingoriginal source · James E. Brennan · Journal of the London Mathematical Society · 1978-10 · DOI 10.1112/jlms/s2-18.2.261 · accessed Aug 15, 2026
2
Brennan's conjecture for semigroups of holomorphic functionsauthoritative webpage · Athanasios Kouroupis · Politecnico di Milano, Geometry, Algebra and their Applications · 2024-06-19; page updated 2026-06-10 · accessed Aug 15, 2026

Important qualifications

Current status is taken from an official university research-seminar page rather than a maintained formal problem registry.
The original publisher page was used for bibliographic identity; no submitted-packet URL was fetched.
No source-package attachment was executed or rendered, and no internal numerical evidence was independently reproduced.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Brennan's Conjecture

Exact mathematical statement

Problem at a glance

Where work on Brennan's Conjecture stands

Brennan's Conjecture in numbers

Prove or refute the augmented boundary-pole lemma for the realizable analytic family, with exact compact and large-pole control.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formalization opportunities

What changed in the research record

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier