lean_workbook_plus_31169

$\frac{(y+z)^2(z+x)}{(2x+y+z)^2}\ge \frac{zy}{x+y}\Leftrightarrow \frac{(x+y)(z+x)}{(2x+y+z)^2}\ge \frac{zy}{(y+z)^2}\Leftrightarrow \frac{(2x+y+z)^2}{(x+y)(z+x)}\leq \frac{(y+z)^2}{yz}\Leftrightarrow \frac{(2x+y+z)^2-4(x+y)(z+x)}{(x+y)(z+x)}\leq \frac{(y+z)^2-4yz}{yz}\Leftrightarrow \frac{(y-z)^2}{(x+y)(z+x)}\leq \frac{(y-z)^2}{yz}\Leftrightarrow (y-z)^2(\frac{x^2+xy+xz}{(x+y)(z+x)yz})\ge 0$