In this paper we obtain a uniqueness theorem of a meromorphic function and its kth derivative that share one small function. So we give some results which are related to a conjecture of R. Brück, and improve some results given by Zhang and Yang (Ann. Acad. Sci. Fenn., Ser. A 1 Math., 32:141–149, 2007) and also answer the open questions posed by Yu (JIPAM. J. Inequal. Pure Appl. Math., 4:1–7, 2003). Furthermore, examples are provided to show that our results are sharp.