\[\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\leq R_{G}\left(x,y,z\right)\leq\min%
\left(\sqrt{\frac{x+y+z}{3}},\frac{x^{2}+y^{2}+z^{2}}{3\sqrt{xyz}}\right).\]