All roots of polynomial inside the open unit disc

Question

I know from here that for a polynomial $p(z)=a_0+a_1z+...+a_nz^n$ with $0<a_0\leq a_1\leq...\leq a_n$ all roots are in the closed unit disk.

What condition do we need to get that all roots are in the open unit disc? I was thinking that maybe some $a_i\neq a_{i+1}$. But I don't know how to prove that?

Answer 1

Here is a sufficient condition: Assume that $0 < a_0 < \dots < a_n$. Set $\tilde p(z) = p(rz)$. Then for $r < 1$, sufficiently close to 1, $\tilde p$ satisfies the conditions in your original post and hence all zeroes of $\tilde p$ are in the unit disk, implying that all zeroes of $p$ are in the disk with radius $r$.

Answer 2

The result you mention is known as the Eneström-Kakeya theorem. Necessary and sufficient conditions for when the roots of the polynomial lie on the boundary of the region are given by Anderson, Saff, and Varga in the paper

N. Anderson, E. B. Saff, and R. S. Varga, On the Eneström-Kakeya theorem and its sharpness, Linear Algebra Appl. 28 (1979), 5-16.

The paper is freely available from Varga's website here.