if X+y+z=1,

minimum value of xy(x+y)^2+yz(y+z)/^2+zx(z+x)^2

wherex,y,z are positive real numbers.

The expression can be made as small as we please. If e is a very small positive number let x=y=e, then z=1-2e. Then the expression is of the order of e. All we can say is that the lower bound is zero