In this paper, we study the uniqueness problems on meromorphic function and its k th order derivative. The results in this paper improve the results given by K. W. Yu (On entire and meromorphic functions that share small functions with their derivatives, J. Inequal. Pure Appl. Math. 4(1)(2003), Art. 21), L. P. Liu and Y. X. Gu (Uniqueness of meromorphic functions that share one small function with their derivatives, Kodai Math. J. 27(2004), 272-279) and supple- ment a result of S. H. Lin and W. C. Lin (Uniqueness of meromorphic functions concerning weakly weighted sharing, Kodai Math. J. 29(2006), 269-280).
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