The angular distribution of mass by Bergman functions

Let D = {z : \z\ < 1} be the unit disk in the complex plane and denote by d A two-dimensional Lebesgue measure on D. For epsilon > 0 we define Sigma(ep silon) = {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-integrab le on D, and f(0) = 0, then integral(f-1(Sigma epsilon)) \f\dA>delta integral(D) \f\dA. This problem arose in connection with a characterization by Hamilton, Reich and Strebel of extremal dilatations for quasiconformal homeomorphisms of D .