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!



Elementary inhomogeneous inequality for three non-negative reals
Denote $x^2=a^3,y^2=b^3,z^2=c^3$. By AM-GM we have $1+2xyz=1+(abc)^{3/2}+(abc)^{3/2}\geqslant 3\sqrt[3]{1\cdot (abc)^{3/2}\cdot (abc)^{3/2}}=3abc$, so LHS is not less then $$a^3+b^3+c^3+3abc\geqslant ab(a+b)+bc(b+c)+ac(a+c)\\ \geqslant 2(ab)^{3/2}+2(bc)^{3/2}+2(ca)^{3/2}=2(xy+yz+zx),$$
the first inequality is [Schur](https://en.wikipedia.org/wiki/Schur%27s_inequality), the second is three AM-GM's.


