Ballieu's Theorem

For any set $\boldsymbol{\mu}=(\mu_1, \mu_2, \ldots, \mu_n)$ of Positive numbers with $\mu_0=0$ and

$$M_\mu=\max_{0\leq k\leq n-1} {\mu_k+\mu_n\vert b_{n-k}\vert\over \mu_{k+1}}.$$

Then all the Eigenvalues $\lambda$ satisfying $P(\lambda)=0$, where $P(\lambda)$ is the Characteristic Polynomial, lie on the Disk $\vert z\vert\leq M_\mu$.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1119, 1979.