- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:43
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
15
1999
1
The angular distribution of mass by Bergman functions
Donald
Marshall
University of Washington, SEATTLE, UNITED STATES
Wayne
Smith
University of Hawai‘i at Mānoa, HONOLULU, UNITED STATES
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 homeo morphisms of $\mathbb D$.
General
93
116
10.4171/RMI/251
http://www.ems-ph.org/doi/10.4171/RMI/251