Normal families of meromorphic functions sharing one function

Suppose p(z) is a holomorphic function, the multiplicity of its zeros is at most d, P(z) is a nonconstant polynomial. Let F be a family of meromorphic functions in a domain D, all of whose zeros and poles have multiplicity at least max{k/2 + d + 1, k + d}. If for each pair of functions f and g in F, P(f)f((k)) and P(g)g((k)) share a holomorphic function p(z), then F is normal in D. It generalizes and extends the results of Jiang, Gao and Wu, Xu.