Normality Criteria of Meromorphic Functions Sharing a Holomorphic Function

Take three integers m≥ 0, k≥ 1 , and n≥ 2 . Let a (≢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∈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^nf^(k) and g^ng^(k) share a (ignoring multiplicity), then F is normal.