Normality Criteria of Meromorphic Functions Sharing a Holomorphic Function

Take three integers m >= 0, k >= 1, and n >= 2. Let a (not equivalent to 0) be a holomorphic function in a domain D of C such that multiplicities of zeros of a are at most m and divisible by n + 1. In this paper, we mainly obtain the following normality criterion: Let F be the family of meromorphic functions on D such that multiplicities of zeros of each f is an element of F are at least k + m and such that multiplicities of poles of f are at least m + 1. If each pair (f, g) of F satisfies that f(n)f(k) and g(n)g(k) share a (ignoring multiplicity), then F is normal.