Revista Matemática Iberoamericana


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Volume 15, Issue 1, 1999, pp. 93–116
DOI: 10.4171/RMI/251

Published online: 1999-04-30

The angular distribution of mass by Bergman functions

Donald E. Marshall[1] and Wayne Smith[2]

(1) University of Washington, Seattle, USA
(2) University of Hawai‘i at Mānoa, Honolulu, USA

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 $\mathbb D$.

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Marshall Donald, Smith Wayne: The angular distribution of mass by Bergman functions. Rev. Mat. Iberoam. 15 (1999), 93-116. doi: 10.4171/RMI/251