Uniqueness of certain power of a meromorphic function sharing a set with its differential monomial

In this paper we are mainly devoted to find out the specific form of a meromorphic function when it shares a set of small functions with its differential monomial counterpart. Our results will improve and extend some of the recent results due to Zhang–Yang [J. L. Zhang and L. Z. Yang, A power of a meromorphic function sharing a small function with its derivative, Ann. Acad. Sci. Fenn. Math. 34(2009), 249–260] and Xu–Yi–Yang [H. Y. Xu, C. F. Yi and H. Wang, On a conjecture of R. Bruck concerning meromorphic function sharing small functions, Revista de Matematica Teoria y Aplicaciones, 23(1)(2016), 291–308]. We provide some examples to show that certain conditions used in the paper can not be removed.