Uniqueness for meromorphic functions and differential polynomials

In this article, we deal with the uniqueness problems on meromorphic functions concerning differential polynomials and prove the following result: Let f and g be two transcendental meromorphic functions, alpha be a meromorphic function such that T(r, α) = o(T(r, f) + T(r, g)) and α notequiv 0,∞.. Let a be a nonzero constant. Suppose that m,n are positive integers such that n m+10. If Ψf and Ψg share (0,2), then (i) if m ≥ 2, then f(z) ≡ g(z) ; (ii) if m = 1, either f(z) ≡ g(z) or f and g satisfy the algebraic equation R(f,g)equiv 0, where R(varpi_1,varpi_2)=(n+1)(varpi_1^{n+2}-varpi_2^{n+2})-(n+2)(varpi_1^{n+1} -varpi_2^{n+1}). The results in this paper improve the results of Xiong-Lin-Mori 14 and the author 12.