Uniqueness polynomials for complex meromorphic functions

A polynomial P(X) in C[X] is called a strong uniqueness polynomial for meromorphic functions if whenever there exist two non-constant meromorphic functions f and g and a complex non-zero constant c such that P(f) = cP(g), then we must have f = g. In this paper, we give a necessary and sufficient condition for a polynomial to be a strong uniqueness polynomial for meromorphic functions under the assumption that P(X) is injective on the roots of P'(X) = 0.