A Schwarz lemma for multivalued functions and distortion theorems for bloch functions with branch points

We give a version of the Schwarz lemma for multivalued mappings between hyp erbolic plane regions. As in the original work of Nehari on this subject, t he derivative must remain bounded near the branch points. Our version of th e distance-decreasing principle represents a considerable strengthening of previous results. We apply it to the study of Bloch functions with branch p oints of specified order. We obtain upper and lower estimates for \f'\, an upper estimate for \f\, and a lower estimate for the radius of the largest schlicht disk in the image of f centered at f(0). We also obtain some resul ts requiring estimates of second order derivatives of f.