Add new constant C_{67} for Brennan's conjecture exponent to README

Author: damek

README.md

@@ -95,6 +95,7 @@ We are arbitrarily numbering the constants as $C_{1}$, $C_{2}$, etc., mostly bas
 | [64](https://teorth.github.io/optimizationproblems/constants/64a.html) | Gauss circle problem exponent | 0 | $\frac{131}{208}$ |
 | [65](https://teorth.github.io/optimizationproblems/constants/65a.html) | Linnik's constant | 1 | 5 |
 | [66](https://teorth.github.io/optimizationproblems/constants/66a.html) | Elliott-Halberstam level-of-distribution exponent | $1/2$ | 1 |
+| [67](https://teorth.github.io/optimizationproblems/constants/67a.html) | Brennan's conjecture exponent | 3.422 | 4 |
 
 
 ## Recent progress

constants/67a.md

@@ -0,0 +1,89 @@
+# 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
+
+$$
+\int_{\Omega}\lvert \varphi'(z)\rvert^p\,dx\,dy\ <\ \infty
+\qquad\text{whenever } \frac{4}{3}<p<4.
+$$
+
+<a href="#HC2015-conjecture-range">[HC2015-conjecture-range]</a>
+
+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
+
+$$
+B_{\mathrm{Bre}}\ >\ 3.
+$$
+
+<a href="#HC2015-brennan-p0">[HC2015-brennan-p0]</a>
+
+The same historical summary attributes the stronger threshold $p_{0}>3.422$ to Bertilsson, so
+
+$$
+B_{\mathrm{Bre}}\ >\ 3.422.
+$$
+
+<a href="#HC2015-best-known-3-422">[HC2015-best-known-3-422]</a>
+
+The conjectural endpoint is
+
+$$
+B_{\mathrm{Bre}}\ =\ 4.
+$$
+
+<a href="#HC2015-conjecture-range">[HC2015-conjecture-range]</a>
+
+Hence the best established range currently is
+
+$$
+3.422\ \le\ C_{67}\ \le\ 4.
+$$
+
+## Known upper bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $4$ | [[HC2015](#HC2015)] | Conjectured endpoint. <a href="#HC2015-conjecture-range">[HC2015-conjecture-range]</a> |
+
+## 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](#HC2015)] | Historical summary attributes this threshold to Bertilsson's dissertation. <a href="#HC2015-best-known-3-422">[HC2015-best-known-3-422]</a> |
+
+## Additional comments and links
+
+- [Wikipedia page on Brennan conjecture](https://en.wikipedia.org/wiki/Brennan_conjecture)
+
+## References
+
+- <a id="HC2015"></a>**[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](https://scholar.google.com/scholar?q=A+better+lower+bound+estimation+of+Brennan%27s+conjecture)
+	- <a id="HC2015-conjecture-range"></a>**[HC2015-conjecture-range]**
+	  **loc:** arXiv PDF p.2, Conjecture 1 and sentence below equation (2)
+	  **quote:** "holds true when $p\in \left(\frac{4}{3},4\right)$."
+	- <a id="HC2015-brennan-p0"></a>**[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."
+	- <a id="HC2015-best-known-3-422"></a>**[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."
+	- <a id="HC2015-1978"></a>**[HC2015-1978]**
+	  **loc:** arXiv PDF p.2, Introduction sentence immediately before Conjecture 1
+	  **quote:** "In 1978 Brennan once hypothesized that:"
+
+- <a id="Bre1978"></a>**[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](https://scholar.google.com/scholar?q=James+E+Brennan+The+integrability+of+the+derivative+in+conformal+mapping)
+
+## Contribution notes
+
+Prepared with assistance from ChatGPT 5.2 Pro.