Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, delta a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x (1), aEuro broken vertical bar, x (n) ) a noncentral multilinear polynomial over C. If delta(G(f(r (1), aEuro broken vertical bar, r (n) ))f(r (1), aEuro broken vertical bar, r (n) )) = 0 for all r (1), aEuro broken vertical bar, r (n) a R, then f(x (1), aEuro broken vertical bar, x (n) )(2) is central-valued on R. Moreover there exists a a U such that G(x) = ax for all x a R and delta is an inner derivation of R such that delta(a) = 0.