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on Brennan's conjecture exponent ($C_{67}$)

The statement

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.

[HC2015-conjecture-range]

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.

[HC2015-brennan-p0]

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

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

[HC2015-best-known-3-422]

The conjectural endpoint is

B_{\mathrm{Bre}}\ =\ 4.

[HC2015-conjecture-range]

Hence the best established range currently is

3.422\ \le\ C_{67}\ \le\ 4.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.