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 $0neq ain r$ 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 $xin r$‎.