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 hyperbolic plane regions. As in the original work of Nehari on this subject, the derivative must remain bounded near the branch points. Our version of the distance-decreasing principle represents a considerable strengthening of previous results. We apply it to the study of Bloch functions with branch points 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 results requiring estimates of second order derivatives of $f$.