Some Normality Criteria of Meromorphic Functions

Let f be a nonconstant meromorphic function in the complex plane C. We shall use the standard notations in Nevanlinna’s value distribution theory of meromorphic functions such as T r, f , N r, f , and m r, f see, e.g., 1, 2 . The notation S r, f is defined to be any quantity satisfying S r, f o T r, f as r → ∞ possibly outside a set of E of finite linear measure. Let F be a family of meromorphic functions on a domain D ⊂ C. We say that F is normal in D if every sequence of functions {fn} ⊂ F contains either a subsequence which converges to a meromorphic function f uniformly on each compact subset of D or a subsequence which converges to∞ uniformly on each compact subset of D. See 1, 3 . The Bloch principle 3 is the hypothesis that a family of analytic meromorphic functions which have a common property P in a domainDwill in general be a normal family if P reduces an analytic meromorphic function in the open complex plane C to a constant. Unfortunately the Bloch principle is not universally true. But it is also very difficult to find some counterexamples about the converse of the Bloch principle, and hence it is interesting to study the problem. In 2005, Lahiri 4 proved the following criterion for the normality, and gave a counterexample to the converse of the Bloch principle by using the result.