Recently, people have proved Hayman′ s conjecture on normal family completely which asserts that a family of meromorphic functions is normal if every function f in the family satisfies f′ -af3≠b,where a and b are fixed complex numbers with a≠0,∞ and b≠∞. In this paper, by replacing f′ by f(k)and setting a condition for the order of poles and zeros of f, we obtain a general theorem: a family of meromorphic functions is normal if every function f in the family satisfies f(k)-afxb and has only poles and zeros of order at least land t respectively, where positive integers n, k, l and t satisfies n- 1 - (k + 1 )/l- 1/t＞ 0. This theorem improves and generalizes previous results in this direction due to H. Chen,Y. Gu,X. Hua, ,X. Pang and W. Schwick.%关于正规族的Hayman猜测目前已完全证实,本文考虑把Hayman猜测中的f'换为一般的f(k),得到一个更为一般的结果,由此改进和推广了陈怀惠,顾永兴,华歆厚,庞学诚与W.Schwick的相应结果.
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