, Volume 77, Issue 1, pp 69-104

Schlicht regions for entire and meromorphic functions

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Abstract

Let f:CC be a meromorpMc function. We study the size of the maximal disc inC, with respect to the spherical metric, in which a single-valued branch of f-1 exists. This 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. 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.

Supported by a Heisenberg fellowship of the DFG.
Partially supported by NSF grant DMS-950036 and by the Lady Davis Foundation.