is there a generalization of Muirhead theorem for negatives reals?

Question

Is there a generalization of Muirhead theorem for negatives reals? Because the original theorem is for only non-negative real numbers.

Answer 1

For negative variables it's wrong.

Try $$x^3+y^3\geq x^2y+xy^2.$$ $(3,0)\succ(2,1)$, but the inequality is wrong for $x+y<0$ and $x\neq y$.

Also, what we need to make with the following inequality? $$x^{\sqrt2}+y^{\sqrt2}\geq2x^{\frac{1}{\sqrt2}}y^{\frac{1}{\sqrt2}}.$$

Sometimes it's true for all reals: $$x^2+y^2\geq2xy,$$ but it not comforting.