lean_workbook_plus_67230

prove that \n\n $\left( {x}^{5}+{y}^{5}+{z}^{5} \right) \left( x+y+z \right) \geq \left( {x}^{2}+{y}^{2}+{z}^{2} \right) \left( {x}^{4}+{y}^{4}+{z}^{4} \right)$ \n\n $\frac{8}{3}\,{\frac { \left( {x}^{2}+{y}^{2}+{z}^{2} \right) ^{2} \left( xy+zx+yz \right) }{ \left( y+z \right) \left( z+x \right) \left( x+y \right) }}\geq \left( \sqrt {{x}^{5}}+\sqrt {{y}^{5}}+\sqrt {{z}^{5}} \right) \left( \sqrt {x}+\sqrt {y}+\sqrt {z} \right)$ \n\n $\frac{3}{2}\,\sum{{\frac {{x}^{3} \left( x+y \right) \left( z+x \right) }{{x}^{2}+yz}}}\geq \left( \sqrt {{x}^{5}}+\sqrt {{y}^{5}}+\sqrt {{z}^{5}} \right) \left( \sqrt {x}+\sqrt {y}+\sqrt {z} \right)$