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 $i=1,2,3$ ?
On
A sufficient condition is 1 > b > c > d > 0 , but I do not know, if there is a condition both sufficient and necessary. Look at kakeya-eneström-theorem.
On
You may be interested in the following thorem.
Theorem : If $$P(z)=\sum_{j=0}^{n} {a_j}^{z_j}$$ is a polynomial of degree $n$ such that $$a_n \ge a_{n−1} \ge \cdots \ge {a_1} \ge {a_0} \gt 0,$$ then $P(z)$ has all its zeros in $|z| \le 1$.
This is well known as Enestr¨om–Kakeya theorem. See here (PDF).
On
The following is Theorem 1.4 in the book Periodicities in Nonlinear Difference Equations, by Grove and Ladas.
Consider the third-degree polynomial equation $$ \lambda^3 + a_2\lambda^2+a_1\lambda + a_0 = 0$$
where $a_0$, $a_1$, $a_2$ are real numbers. Then a necessary and sufficeint condition that all roots lie in the open disk $|\lambda| < 1$ is
$$ | a_2 + a_0|<1+a_1, \quad |a_2 - 3a_0|< 3 - a_1,\quad \text{and} \quad a_0^2+a_1+a_0a_2 < 1. $$
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: