Extremal domains for the geometric reformulation of Brennan's conjecture (Q5940619)

One of the equivalent statement of Brennan's conjecture is that for any conformal mapping \(f\) of the unit disk \(|z|<1\) into \(\mathbb{C}\), and \(p\in (-\infty, -2]\) and \(\varepsilon>0\), NEWLINE\[NEWLINE\int^{2\pi}_0 |f^\prime (re^{it})|^p dt =O\left( \frac{1}{(1-r)^{|p|-1+\varepsilon}}\right),\quad r\to 1.NEWLINE\]NEWLINE The author discusses some other formulation of this conjecture, especially some inequalities for so-called beta numbers. It is proved in the paper that the extremal domain in the mentioned inequality has a boundary with special property. The second variation of Chang, Schiffer and Schober was the main tool.