Abstract: Let F be a family of meromorphic functions in a domain D,and let k,n(n≥k+2) be  positive integers.Let a≠0 be a finite complex number.If for each f∈F,the zeros of f have multiplicity at least  n,and G(f) and G(g) share b in D for every pair of functions f,g∈F,where G(f)=P(f(k))+H(f) is a differential polynomial of f Fis normal in D.

Key words: meromorphic functions, normal family, differential polynomial