# The angular distribution of mass by Bergman functions

### Donald E. Marshall

University of Washington, Seattle, USA### Wayne Smith

University of Hawai‘i at Mānoa, Honolulu, USA

## 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 $\epsilon > 0$ we define $\sum_\epsilon = z:|$ arg $z | < \epsilon$. We prove that for every $\epsilon > 0$ there exists a $\delta > 0$ such that if $f$ is analytic, univalent and area-integrable on $\mathbb D$, and $f(0) = 0$, then

$\int _{f^–1(\sum_\epsilon)} | f | d\mathcal A > \delta \int_\mathbb D | f | d\mathcal A$

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

## Cite this article

Donald E. Marshall, Wayne Smith, The angular distribution of mass by Bergman functions. Rev. Mat. Iberoam. 15 (1999), no. 1, pp. 93–116

DOI 10.4171/RMI/251