Locally univalent functions, VMOA and the Dirichlet space

We study geometric properties of the image of the unit circle under a bounded locally univalent function g such that log g' belongs either to the Dirichlet space D, VMOA or the little Bloch space B-0. Concerning VMOA and B-0, our findings generalize the corresponding results for conformal maps shown by Pommerenke in the late 1970s. In the case of D, we give a strictly geometric necessary condition for g to satisfy log g'is an element of D, and also offer two different 'semi-geometric' characterizations of when log g'is an element of D