Koebe sequences of arcs and normal meromorphic functions

Abstract:

Let f be a normal meromorphic function in the unit disk. An estimate for the growth of the modulus of f on a Koebe sequence of arcs is obtained; the estimate is in terms of the order of normality of f. An immediate consequence of the estimate is the following theorem due to F. Bagemihl and W. Seidel: A nonconstant normal meromorphic function has no Koebe values. Another consequence is that each level set of a nonconstant normal meromorphic function cannot contain a Koebe sequence of arcs provided the order of normality of f is less than a certain positive constant ${C^\ast }$.