**Abstract** J. E. Brennan in 1978 conjectured that if $f:\mathbb{D}\to\mathbb{C}$ is a conformal map, then the $p$−integral means of the derivative are finite, whenever $p \in (-2,\frac{2}{3})$.
That is:

Brennan’s conjecture is one of the most famous remaining open problems in the field of geometric function theory. It is known that the conjecture holds for the values $p \in (−1.752, \frac{2}{3})$. The aim of this talk is to give a short proof of Brennan’s conjecture in the special case where f can be embedded into a non-elliptic continuous semigroups of holomorphic functions in the unit disk.

This is joint work with Alexandru Aleman.