Plot[{((z (Coth[z] + Coth[z]) + Sqrt[(z^2 (Coth[z] + Coth[z])^2) -
         4 (0.5 + Coth[z]*Coth[z]) z^2 (0.5)])/(2 (0.5 + Coth[z]*Coth[z]))),
       ((z (Coth[z] + Coth[2 z]) + Sqrt[(z^2 (Coth[z] + Coth[2 z])^2) - 4 (0.9 + Coth[z]*Coth[2 z]) z^2 (0.9)])/(2 (0.9 + Coth[z]*Coth[2 z])))},
{z, 0, 9}, PlotLegends -> "ExpressFrame -> True , FrameLabel -> {[Xi]h, [Omega]}]