Some specific cases of linear preservers of rsgut-majorization on R5

A (not necessarily nonnegative) real matrix $R$ with the property $Re\leq e$ is said to be a $\textit{g-row substochastic}$ matrix, where $e=(1,\ldots,1)^t$. For $x,y\in \mathbb{R}_{n}$ , we say that $x$ rsgut-majorized by $y$ (write as $x\prec_{rsgut}y$) if for some g-row substochastic matrix $R$ $x=yR$. Here, we give an equivalent condition for rsgut-majorization on $\mathbb{R}_{5}$. Also, we characterize some specific cases of this relation on $\mathbb{R}_{5}$.