Schlicht regions for entire and meromorphic functions

Let f : C → C be a meromorphic function. We study the size of the maximal disc in C with respect to spherical metric, in which a singlevalued branch of f−1 exists. The problem is related to normality and type criteria. Best possible lower estimates of the size of such discs are obtained for entire functions and a class of meromorphic functions containing all elliptic functions. An estimate for the class of rational functions is also given which is best possible for rational functions of degree 7. We do not know whether it is asymptotically best possible when the degree tends to infinity. For algebraic functions of given genus we obtain an estimate which is precise for genera 2 and 5 and asymptotically best possible when the genus tends to infinity.