lean_workbook_plus_30161

Another way also using Holder: \n${\frac { \left( x \left( {x}^{2}+4\,xy \right) \left( x+3\,y+7\,z \right) +y \left( {y}^{2}+4\,yz \right) \left( y+3\,z+7\,x \right) +z \left( 4\,xz+{z}^{2} \right) \left( z+3\,x+7\,y \right) \right) ^{3}}{x \left( {x}^{2}+4\,xy \right) ^{2} \left( x+3\,y+7\,z \right) ^{3}+y \left( {y}^{2}+4\,yz \right) ^{2} \left( y+3\,z+7\,x \right) ^{3}+z \left( 4\,xz+{z}^{2} \right) ^{2} \left( z+3\,x+7\,y \right) ^{3}}} \geq \frac{5}{9}\, \left( x+y+z \right) ^{4}$