Uniqueness and value-sharing of meromorphic functions

Abstract. Concerning the uniqueness and sharing values of meromorphic functions, many results about meromorphic functions that share more than or equal to two values have been obtained. In this paper, we shall study meromorphic functions that share only one value, and prove the following result: For n ≥ 11 and two meromorphic functions f(z) and g(z) , if ff ′ and gg share the same nonzero and finite value a with the same multiplicities, then f ≡ dg or g = c1e cz and f = c2e −cz , where d is an (n + 1)th root of unity, c , c1 and c2 being constants. As an application, we solve some non-linear differential equations.