I was working on an inequality proof in which I need to use the following inequality to conclude.
$$\forall~a,b,c\gt 0~,~a^2+b^2+c^2=(abc)^2-2\implies a^2+b^2+c^2\leq 6$$
I can't think of any way to prove this. I only need a hint since I dislike people just giving away the answers. Thanks in advance!
If this claim is false, I'd like a counter-example. Thanks again!