A product of two generalized derivations on polynomials in prime rings

Let $R$ be a prime ring of characteristic different from 2, let $F$ and $G$ be non-zero generalized derivations of $R$ and let $f(x_1, \ldots, x_n)$ a polynomial. Let $f(R)$ be the set $\{f(r_1, \ldots, r_n) \mid r_1, \ldots, r_n\in R \}$. The purpose of this paper is to study the situation when the composition $(FG)$ acts on the elements of $f(R)$ as a generalized derivation on the elements of $f(R)$ then $(FG)$ is a generalized derivation of $R$ and one of the following holds:
\begin{enumerate}
\item there exists $\alpha\in C$ such that $F(x)=\alpha x$ for all $x\in R$;
\item there exists $\alpha\in C$ such that $G(x)=\alpha x$ for all $x\in R$;
\item there exists $\alpha, b \in U$ such that $F(x)=ax,G(x)=bx$, for all $x\in R$;
\item there exists $\alpha, b \in U$ such that $F(x)=xa,G(x)=xb$, for all $x\in R$;
\item there exists $a, b \in U, \alpha,\beta\in C$ such that $F(x)=ax+xb$, $G(x)=\alpha x+\beta(ax-xb)$, for all $x\in R$.
\end{enumerate}