19 Giugno 2024 17:00 CEST Zoom online
Athanasios Kouroupis Norwegian University of Science and Technology, Trondheim

Brennan's conjecture for semigroups of holomorphic functions

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:

$\displaystyle \int_{\mathbb{D}} \vert f'(w)\vert^p\, \mathrm{d}A(w) < \infty,\quad p \in (-2,\frac{2}{3})$

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.

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