Conformal contractions and lower bounds on the density of harmonic   measure
  We give a concrete sufficient condition for a simply-connected domain to be the image of the unit disk under a nonexpansive conformal map. This class of domains is also characterized by having sufficiently dense harmonic measure. The relation with the harmonic measure provides a natural higher-dimensional analogue of this problem, which is also addressed. 
Leonid V. Kovalev
math.CV