Uniqueness and value-sharing of meromorphic functions

In this paper, we study the uniqueness of meromorphic functions and prove the following theorem: Let f(z)f(z) and g(z)g(z) be two non-constant meromorphic functions, n,kn,k two positive integers with n>3k+8n>3k+8. If [fn(z)](k)[fn(z)](k) and [gn(z)](k)[gn(z)](k) share 1 CM, then either f(z)=c1ecz,g(z)=c2e−cz, where c1,c2c1,c2 and cc are three constants satisfying (−1)k(c1c2)n(nc)2k=1(−1)k(c1c2)n(nc)2k=1 or f(z)=tg(z)f(z)=tg(z) for a constant tt such that tn=1tn=1. Our results improves the results of Fang [M.L. Fang, Uniqueness and value-sharing of entire functions, Comput. Math. Appl. 44 (2002) 823–831. [7]], Fang and Hong [M.L. Fang, W. Hong, A unicity theorem for entire functions concerning differential polynomials, Indian J. Pure Appl. Math. 32 (9) (2001) 1343–1348. [8]] and Lin and Yi [W.-C. Lin, H.-X. Yi, Uniqueness theorems for meromorphic function, Indian J. Pure Appl. Math. 32 (9) (2004) 121–132. [9]].