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


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

f1(ϵ)fdA>δDfdA\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 D\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