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Brennan's conjecture exponent (C67C_{67})

Brennan's conjecture exponent: determine C67C_{67} or improve its bounds; the repository records 3.422 (lower) and 4 (upper).

MSC 30C35 · General theory of conformal mappings

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let Ω⊂C\Omega\subset\mathbb{C} be simply connected with at least two boundary points in the extended complex plane, and let φ:Ω→D\varphi:\Omega\to\mathbb{D} be a conformal map. Brennan's conjecture states that

∫Ω∣φ′(z)∣p dx dy < ∞whenever 43<p<4.\int_{\Omega}\lvert \varphi'(z)\rvert^p\,dx\,dy\ <\ \infty \qquad\text{whenever } \frac{4}{3}<p<4.

[HC2015-conjecture-range]

We define

C67 := BBre,C_{67}\ :=\ B_{\mathrm{Bre}},

where BBreB_{\mathrm{Bre}} is the supremum of exponents pp for which the above integrability statement holds for all such Ω\Omega and φ\varphi.

Brennan proved the range 4/3<p<p04/3<p<p_{0} for some p0>3p_{0}>3, so

BBre > 3.B_{\mathrm{Bre}}\ >\ 3.

[HC2015-brennan-p0]

The same historical summary attributes the stronger threshold p0>3.422p_{0}>3.422 to Bertilsson, so

BBre > 3.422.B_{\mathrm{Bre}}\ >\ 3.422.

[HC2015-best-known-3-422]

The conjectural endpoint is

BBre = 4.B_{\mathrm{Bre}}\ =\ 4.

[HC2015-conjecture-range]

Hence the best established range currently is

3.422 ≤ C67 ≤ 4.3.422\ \le\ C_{67}\ \le\ 4.

Lean 4

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Background

References (from the repository page)

  • [HC2015] Hu, Junyi; Chen, Shiyu. A better lower bound estimation of Brennan's conjecture. arXiv:1509.00270 (2015). DOI: https://doi.org/10.48550/arXiv.1509.00270. arXiv PDF: https://arxiv.org/pdf/1509.00270.pdf. Publisher: https://arxiv.org/abs/1509.00270. Google Scholar

  • [HC2015-conjecture-range] loc: arXiv PDF p.2, Conjecture 1 and sentence below equation (2) quote: "holds true when p∈(43,4)p\in \left(\frac{4}{3},4\right)."

  • [HC2015-brennan-p0] loc: arXiv PDF p.2, Introduction, item 3 in the historical list quote: "Brennan [3] proved that p∈(43,p0)p\in \left(\frac{4}{3},p_{0}\right) (p0>3)(p_{0}>3) holds true."

  • [HC2015-best-known-3-422] loc: arXiv PDF p.2, Introduction, item 4 in the historical list quote: "Bertilsson [1] dissertation, KTH Sweden, 1990 proved that (p0>3.422)(p_{0}>3.422) and this is the most promising result obtained so far."

  • [HC2015-1978] loc: arXiv PDF p.2, Introduction sentence immediately before Conjecture 1 quote: "In 1978 Brennan once hypothesized that:"

  • [Bre1978] Brennan, James E. The integrability of the derivative in conformal mapping. Journal of the London Mathematical Society (2) 18 (1978), no. 2, 261-272. DOI: https://doi.org/10.1112/jlms/s2-18.2.261. Google Scholar

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 3.422; best known upper bound 4. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
44 [HC2015] Conjectured endpoint. [HC2015-conjecture-range]

Known lower bounds

Bound Reference Comments
22 Trivial by change of variables: ∫Ω∣φ′(z)∣2 dA(z)=Area(D)=π\int_{\Omega}\lvert\varphi'(z)\rvert^2\,dA(z)=\mathrm{Area}(\mathbb{D})=\pi.
3.4223.422 [HC2015] Historical summary attributes this threshold to Bertilsson's dissertation. [HC2015-best-known-3-422]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Brennan's conjecture exponent ($C_{67}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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