Find the smallest number $\alpha$, such that for all $x,y,z$ $\alpha(x^2-x+1)(y^2-y+1)(z^2-z+1)\ge(xyz)^2-xyz+1$ Find a smallest number $\alpha$, such that for all $x,y,z$ (not all of which are positive) inequality
$$\alpha(x^2-x+1)(y^2-y+1)(z^2-z+1)\ge(xyz)^2-xyz+1$$

My work so far:
Let $f(t)=t^2-t+1$. Then $f(t) \ge \frac34$.
If $x=0, y=z=\frac12$, then $$\alpha\ge \frac{16}9$$
In the starting formulation the answer is $\frac{16}{9}$.

If $x$, $y$ and $z$ are non-positives so after replacing $x\rightarrow-x$...
we need to prove that 
$$\frac{16}{9}(x^2+x+1)(y^2+y+1)(z^2+z+1)\geq x^2y^2z^2-xyz+1$$

which is true for all non-negatives $x$, $y$ and $z$ because we'll prove now that even
$$\frac{16}{9}(x^2+x+1)(y^2-y+1)(z^2-z+1)\geq x^2y^2z^2+xyz+1$$.

If $x\leq0$, $y\leq0$ and $z\geq0$ we need to prove that
$$\frac{16}{9}(x^2+x+1)(y^2+y+1)(z^2-z+1)\geq x^2y^2z^2-xyz+1$$ 
for non-negatives $x$, $y$ and $z$, which follows from
$$\frac{16}{9}(x^2+x+1)(y^2-y+1)(z^2-z+1)\geq x^2y^2z^2+xyz+1$$
again.

Now we'll prove it:

$$\begin{align}
&amp;16(x^2+x+1)(y^2-y+1)(z^2-z+1)-9(x^2y^2z^2+xyz+1)
\\&amp;\phantom{aaa}=(16(y^2-y+1)(z^2-z+1)-9y^2z^2)x^2+(16(y^2-y+1)(z^2-z+1)-9yz)x
\\
&amp;\phantom{aaaaa}+16(y^2-y+1)(z^2-z+1)-9
\\&amp;\phantom{aaa}
\geq\left(16\left(\frac{3}{4}y^2+\left(\frac{y}{2}-1\right)^2\right)\left(\frac{3}{4}z^2+\left(\frac{z}{2}-1\right)^2\right)-9y^2z^2\right)x^2+
+(3y\cdot3z-9yz)x
\\&amp;\phantom{aaaaa}+
16\left(\frac{3}{4}+\left(\frac{1}{2}-y\right)^2\right)\left(\frac{3}{4}+\left(\frac{1}{2}-z\right)^2\right)-9
\\&amp;\phantom{aaa}\geq0
\end{align}$$
From your
$a(x^2-x+1)^3\ge x^6-x^3+1
$,
since
$(x+1)(x^2-x+1)
=x^3+1
$
and
$(x^3+1)(x^6-x^3+1)
=x^8+1
$,

$a(\frac{x^3+1}{x+1})^3\ge \frac{x^9+1}{x^3+1}
$
or
$a\ge \frac{(x^9+1)(x+1)^3}{(x^3+1)^4}
$.

According to Wolfy,
this has a maximum of
$2.1547$
at $x \approx 0.435421
$
and
$2.29663
$.
These are the roots of
$0
= x^6-2 x^5-x^4+x^2+2 x-1
$.

The exact roots are
$-1, 1,
\frac12 (1\pm\sqrt{2} 3^{1/4}+\sqrt{3}),
\frac12 (1\pm i\sqrt{2} 3^{1/4}-\sqrt{3})
$.

So $a$ must be at least
$2.1547$.