Abstract: Let k be a positive integer,let h（z）■0 be a holomorphic functions in a domain D,and let F be a family of zero-free meromorphic functions in D,all of whose poles have order at least l.If,for each f∈P（f）（z）-h（z） has at most k+l-1 distinct zeros（ignoring multiplicity） in D,where P（f）（z）=f^（k）（z）+a₁（z）f⁽（k-1）（z）+…+ak（z）f（z） is a differential polynomial of f and aj（z）（j=1,2,···,k） are holomorphic functions in D,then F is normal in D. 

Key words: Meromorphic Functions, Normality, Zero-free