<s>[INST] Absolute value of cubic polynomial roots lower than 1 Assume we have a cubic polynomial 
$ x^3 +bx^2+xc+d=0 $, with $b,c,d$ real numbers.
Let $x_1, x_2, x_3 $ be the roots, either real or complex.
What is the relation of the coefficients $b,c$ and $d$ in order to have the roots inside the unit sphere, that means
$ |x_i| < 1$  for  $i=1,2,3$ ? 
 [/INST] What you're trying to determine is whether the conditions on $b,c,d$ that make $x^3 + bx^2 + cx + d$ a Schur polynomial. As mentioned before, a sufficient (but perhaps unnecessary) condition is that
$$
1>b>c>d
$$
Is true.  For the precise conditions, one may apply either the Jury test or Bistritz test
Or, apply the Routh-Hurwitz criterion to 
$$
(z-1)^3p\left(\frac{z+1}{z-1}\right) = \\(1+b+c+d)z^3+ 
(3+b-c-3d)z^2 + (3 - b - c - 3d)z + (1 - b + c - d)
$$
That last option leads to the following statement:
Let $a_3 = 1+b+c+d$, $a_2 = 3+b-c-3d$, $a_1 = 3 - b - c - 3d$, and $a_0 = 1 - b + c - d$. We may state that $x^3 + bx^2 + cx + d$ has its roots in the unit ball if and only if all of the following conditions are satisfied:


*

*$a_i>0$ for each $i$

*$a_2 a_1> a_3 a_0$

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