The angular distribution of mass by Bergman functions

Abstract

Let $\mathbb{D}=z:|z|<1$ be the unit disk in the complex plane and denote by $d\mathcal{A}$ two-dimensional Lebesgue measure on $\mathbb{D}$. For $ϵ>0$ we define ${\sum }_{ϵ}=z:|$ arg $z|<ϵ$. We prove that for every $ϵ>0$ there exists a $\delta >0$ such that if $f$ is analytic, univalent and area-integrable on $\mathbb{D}$, and $f(0)=0$, then

. This problem arose in connection with a characterization by Hamilton, Reich and Strebel of extremal dilatations for quasiconformal homeomorphisms of $\mathbb{D}$.