Is there a relation of roots for scaled coefficients of real monic polynomial?

Question

Let $\pi$ be the bijection between coefficients of the real monic polynomials to the real monic polynomials. Let $a\in \mathbb R^n$ be fixed vector. Then \begin{align*} \pi(a) = t^n + a_{n-1} t^{n-1} + \dots + a_0. \\ \end{align*} Let $r > 0$. Is it possible to bound the roots of $\pi(ra)$ using the roots $\pi(a)$ in the sense: if the roots of $\pi(a)$ are in the unit disk, then the roots of $\pi(ra)$ would be contained in the unit disk for some suitable choice of $r > 0$? I tried to use Vieta's formula, but it gets quite complicate.