Brennan’s conjecture exponent
Description of constant
Let $\Omega\subset\mathbb{C}$ be simply connected with at least two boundary points in the extended complex plane, and let $\varphi:\Omega\to\mathbb{D}$ be a conformal map. Brennan’s conjecture states that
\[\int_{\Omega}\lvert \varphi'(z)\rvert^p\,dx\,dy\ <\ \infty \qquad\text{whenever } \frac{4}{3}<p<4.\]We define
\[C_{67}\ :=\ B_{\mathrm{Bre}},\]where $B_{\mathrm{Bre}}$ is the supremum of exponents $p$ for which the above integrability statement holds for all such $\Omega$ and $\varphi$.
Brennan proved the range $4/3<p<p_{0}$ for some $p_{0}>3$, so
\[B_{\mathrm{Bre}}\ >\ 3.\]The same historical summary attributes the stronger threshold $p_{0}>3.422$ to Bertilsson, so
\[B_{\mathrm{Bre}}\ >\ 3.422.\]The conjectural endpoint is
\[B_{\mathrm{Bre}}\ =\ 4.\]Hence the best established range currently is
\[3.422\ \le\ C_{67}\ \le\ 4.\]Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $4$ | [HC2015] | Conjectured endpoint. [HC2015-conjecture-range] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $2$ | Trivial by change of variables: $\int_{\Omega}\lvert\varphi’(z)\rvert^2\,dA(z)=\mathrm{Area}(\mathbb{D})=\pi$. | |
| $3.422$ | [HC2015] | Historical summary attributes this threshold to Bertilsson’s dissertation. [HC2015-best-known-3-422] |
Additional comments and links
References
- [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 $p\in \left(\frac{4}{3},4\right)$.”
- [HC2015-brennan-p0] loc: arXiv PDF p.2, Introduction, item 3 in the historical list quote: “Brennan [3] proved that $p\in \left(\frac{4}{3},p_{0}\right)$ $(p_{0}>3)$ holds true.”
- [HC2015-best-known-3-422] loc: arXiv PDF p.2, Introduction, item 4 in the historical list quote: “Bertililsson [1] issertation, KTH Sweden, 1990 proved that $(p_{0}>3.422)$ 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
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.