Normal Criteria for Family Meromorphic Functions Sharing Holomorphic Function

In this paper, we study the value distribution of differential polynomial with the form f(n) (f(n1))((t1))... (f(nk))((tk)) where f is a transcendental meromorphic function. Namely, we prove that f(n) (f(n1))((t1))... (f(nk))((tk)) - P (z) has infinitely zeros, where P(z) is a nonconstant polynomial and n is an element of N, k, n(1),..., n(k), t(1),..., t(k) are positive integer numbers satisfying n + Sigma(k)(u) n(u) >= Sigma(k)(u=1) t(u) + 3. Using it, we establish some normality criterias for family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. Our results generalize some previous results on normal family of meromorphic functions.