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

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‎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 $0neq ain R$ and $f(x_1,ldots,x_n)$ 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 $a(H(f(x))f(x)-f(x)G(f(x)))in C$ for all‎ ‎$x=(x_1,ldots,x_n)in R^n$‎. ‎ one of the following holds‎: ‎$f(x_1,ldots,x_n)^2$ is central valued on $R$ and there exist $b,p,qin U$ such‎ ‎that $H(x)=px+xb$ for all $xin R$‎, ‎$G(x)=bx+xq$ for all $xin R$ with $a(p-q)in C$;‎ ‎there exist $p,qin U$ such that $H(x)=px+xq$ for all $xin R$‎, ‎$G(x)=qx$ for all $xin R$ with $ap=0$;‎  $f(x_1,ldots,x_n)^2$ is central valued on $R$ and there exist $qin U$‎, ‎$lambdain C$ and an outer derivation $g$ of $U$‎ ‎such that $H(x)=xq+lambda x-g(x)$ for all $xin R$‎, ‎$G(x)=qx+g(x)$ for all $xin R$‎, ‎with $ain C$;‎ $R$ satisfies $s_4$ and there exist $b,pin U$ such‎ ‎that $H(x)=px+xb$ for all $xin R$‎, ‎$G(x)=bx+xp$ for all $xin R$‎.