Exceptional functions/values wandering on the sphere and normal families
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.