TITLE: Elementary inhomogeneous inequality for three non-negative reals
QUESTION [5 upvotes]: I need the following estimate for something I am working on, but I don't immediately see how to establish it.
For $x, y, z \in \mathbb{R}_{\ge 0}$, show that
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz + zx),$$
and I suspect the only point of equality is (1,1,1).
It feels like the sort of thing that ought to have a simple Olympiad-style proof using standard inequalities, but I haven't had any luck thus far; of course, I'll take any proof that I can get.
Also, if this inequality is reminiscent of any others, I would be grateful for references!

REPLY [3 votes]: Another way.
Since $$\prod_{cyc}((x-1)(y-1))=\prod_{cyc}(x-y)^2\geq0,$$ we can assume that
$$(x-1)(y-1)\geq0,$$ which gives $$z(x-1)(y-1)\geq0$$ or
$$xyz\geq xz+yz-z.$$
Id est, it's enough to prove that:
$$2xz+2yz-2z+x^2+y^2+z^2+1\geq2(xy+xz+yz)$$ or
$$(x-y)^2+(z-1)^2\geq0$$ and we are done!