Exceptional functions/values wandering on the sphere and normal families

יום ב', 11/11/2013 - 14:00

We extend Caratheodory's generalization of Montel's fundamental normality test
to "wandering" exceptional functions (i.e. depending on the respective function in the
family under consideration), and we give a corresponding result on shared functions.
Furthermore, we prove that if we have a family of pairs (a,b) of functions meromorphic
in a domain such that a and b uniformly "stay away from each other " , then the families
of the functions a resp.  b are normal. The proofs are based on a "simultaneous rescaling"
version of Zalcman's Lemma. We also introduce a somewhat "strange" result about some
sharing wandering values assumptions that imply normality.