Prove or disprove $3(a^2b^2+b^2c^2+a^2c^2)\ge (ab+bc+ac)(a^2+b^2+c^2)$ Let $a,b,c\ge 0$ Prove or disprove
$$3(a^2b^2+b^2c^2+a^2c^2)\ge (ab+bc+ac)(a^2+b^2+c^2)$$
I have try use Cauchy-Schwarz inequality:
$$3(a^2b^2+b^2c^2+c^2a^2)\ge (ab+bc+ca)^2$$
But $ab+bc+ca\le a^2+b^2+c^2$

The given inequality is equivalent to
$$ \sum_{sym}(3a^2 b^2-a^2 bc -a^3 b)\geq 0 $$
that does not hold for $(a,b,c)=(1,2,3)$. However, a similar inequality, namely
$$ \sum_{sym}(a^4-2a^3 b+a^2 bc)\geq 0 $$
holds by Schur's inequality with $t=2$.
