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, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x1, …, xn) a noncentral multilinear polynomial over C. If δ(G(f(r1, …, rn))f(r1, …, rn)) = 0 for all r1, …, rn ∈ R, then f(x1, …, xn)2 is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.