Prove that $xy+yz+zx+\frac{x^2y^2}{z^2}+\frac{y^2z^2}{x^2}+\frac{z^2x^2}{y^2}\ge 2x^2+2y^2+2z^2$ for every $x,y,z$ strictly positive I checked the inequality for $y=z$. In this particular case, after some simplifications, the inequality becomes:
$$
2x^3+y^3\ge 3x^2y,
$$
which is true, according to the arithmetic-geometric mean inequality applied to the numbers $x^3,\ x^3$ and $y^3$. I have no idea, at least for now, on how to proceed in the general case. Please give me a hint.

Use Schur inequality
$$a^4+b^4+c^4+abc(a+b+c)\ge \sum_{cyc}(a^3b+ab^3)\ge \sum_{cyc}2a^2b^2$$
Let
$$a=xy,b=yz,c=xz$$
then is your inequality
