# All roots of polynomial inside the open unit disc

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?

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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 $r$.