Linear functions preserving rsg-majorization on R2

A (not necessarily nonnegative) real matrix $R$ with the property $Re=e$ is said to be a $\textit{g-row stochastic}$ matrix, where $e=(1,\ldots,1)^t$. For $x,y\in \mathbb{R}_{n}$ , we say that $x$ rsg-majorized by $y$ (write as $x\prec_{rsg}y$) if for some symmetric g-row stochastic matrix $R$ with all its main diagonal entries equal, $x=yR$. Here, we give an equivalent condition for rsg-majorization on $\mathbb{R}_{2}$. Also, the possible structures of linear preserving rsg-majorization functions from $\mathbb{R}_{2}$ to $\mathbb{R}_{2}$ are found. Finally, all linear strongly preserving $\prec_{rsg}$ from $\mathbb{R}_{2}$ to $\mathbb{R}_{2}$ are characterized.