Brennan conjecture

Problem's Description

The Brennan conjecture, introduced by James E. Brennan in 1978, [1] is a significant hypothesis within the realm of complex analysis. It revolves around estimating, under certain conditions, the integral powers of the moduli of the derivatives of conformal maps into the open unit disk. Specifically, the conjecture deals with a simply connected open subset $W$ of the complex plane, with at least two boundary points in the extended complex plane, and a conformal map $\psi$ of $ W$ onto the open unit disk. The Brennan conjecture posits that $$\int_W |\psi'|^q\,dxdy < \infty, \qquad \hbox{ whenever } 4/3 < q < 4.$$ While Brennan provided a proof for $4/3 < q < q_0$ for some constant $q_0 > 3$, it remains unresolved for the entire range. In 1999, Bertilsson demonstrated that the conjecture holds true when $4/3 < q < 3.422$, but the complete resolution of the conjecture is yet to be achieved.

See a brief review at this page:

https://www.uma.es/charm2011/talks/02-1930-GStylogiannis.pdf

And the thesis: Bertilsson, Daniel (1999). On Brennan's conjecture in conformal mapping (PDF). Kungliga Tekniska Högskolan; 110 pages,

https://www.diva-portal.org/smash/get/diva2:8593/FULLTEXT01.pdf

For a lower bound see: https://arxiv.org/abs/1509.00270

1. ArticleIs an originThe integrability of the derivative in conformal mapping

   • Brennan,
   • James E.

   Journal of the London Mathematical Society 2, 261–272fulltext