Higher-Dimensional Generalizations of Some Theorems on Normality of Meromorphic Functions

Abstract

In [5], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions F in a domain D⊂C, and for a positive constant ε, if for each f∈F there exist meromorphic functions af,bf,cf such that f omits af,bf,cf in D and min{ρ(af(z),bf(z)),ρ(bf(z),cf(z)),ρ(cf(z),af(z))}≥ε for all z∈D, then F is normal in D. Here, ρ is the spherical metric in Cˆ. In this paper, we establish the high-dimensional versions for the above result and for the following well-known result of Lappan: A meromorphic function f in the unit disc △:={z∈C:|z|<1} is normal if there are five distinct values a1,…,a5 such that sup{(1−|z|2)|f′(z)| 1+|f(z)|2:z∈f−1{a1,…,a5}}<∞.