Co-centralizing generalized derivations acting on multilinear polynomials in prime rings

‎‎Let R be a noncommutative prime ring of‎ ‎characteristic different from 2‎, ‎U the Utumi quotient ring of R‎, ‎C (=Z(U)) the extended centroid‎ ‎of R‎. ‎Let 0≠a∈R and f(x1,…,xn) a multilinear‎ ‎polynomial over C which is noncentral valued on R‎. ‎Suppose‎ ‎that G and H are two nonzero generalized derivations of R‎ ‎such that a(H(f(x))f(x)−f(x)G(f(x)))∈C for all‎ ‎x=(x1,…,xn)∈Rn‎. ‎ one of the following holds‎: ‎(1) f(x1,…,xn)2 is central valued on R and there exist b,p,q∈U such‎ ‎that H(x)=px+xb for all x∈R‎, ‎G(x)=bx+xq for all x∈R with a(p−q)∈C;‎ ‎(2) there exist p,q∈U such that H(x)=px+xq for all x∈R‎, ‎G(x)=qx for all x∈R with ap=0;‎ (3) f(x1,…,xn)2 is central valued on R and there exist q∈U‎, ‎λ∈C and an outer derivation g of U‎ ‎such that H(x)=xq+λx−g(x) for all x∈R‎, ‎G(x)=qx+g(x) for all x∈R‎, ‎with a∈C;‎ (4) R satisfies s4 and there exist b,p∈U such‎ ‎that H(x)=px+xb for all x∈R‎, ‎G(x)=bx+xp for all x∈R‎.