A result concerning b-generalized skew derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, Z(R) its center, Q(r) its right Martindale quotient ring, C its extended centroid, f (x(1), ..., x(n)) a non-central multilinear polynomial over C, b is an element of Q(r) and F, G two b-generalized skew derivations of R. If there exists a is an element of R\Z(R) such that [a, F(f(r(1), ..., r(n))) f(r(1), ..., r(n))] = f (r(1), ...., r(n))G (f(r(1), ..., r(n))) for any r(1), ..., r(n) is an element of R, then f(x(1), ..., x(n))(2) is central valued on R and there exist p, q is an element of Q(r) such that F(x) = px,G(x) = xq for all x is an element of R with [a, p] = q.