Generalized derivations which extend the concept of Jordan homomorphism

Let R be a prime ring, U the right Utumi quotient ring of R, C its extended centroid, I a non-zero right ideal of R, f (x1, . . . , xn) a non-central multilinear polynomial over C, F , G two generalized derivations of R, m 1 a fixed integer. Denote f (I) the set of all evaluations of the polynomial f (x1, . . . , xn) in I. If F (u m ) = G(u) m , for any u f (I), then we describe all possible forms of F and G.