Given a polynomial $p(x)=\sum_k a_k x^k$ with $x,a_k\in \mathbb{R}$, I am looking for a lower bound on roots $r_i$ of the following kind:
$$ \min_i |r_i| \geq ? $$
So far, I found this paper which says (Theorem 2.15): For $p(x) = \sum_{v=0}^n a_v x^v $ with $a_n=1$ and $A=\max_{0\leq v\leq n-1}|a_v|$, $p(x)$ has all its zeroes in the ring-shaped region: $$ \frac{|a_0|}{ 2 (1+A)^{n-1} (An+1) } \leq |x| \leq 1 + \lambda_0 A, $$ where I skipped the $\lambda_0$ part, as I am interested in the left side of the inequality.
The problem with this bound is that it does not seem to be very tight, particularly for high-order polynomials. Therefore, I am asking whether more/tighter bounds are known.