Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far from every root. That is, for every $b$ such that $p(b)=0$, $|a-b| \ge \epsilon$. Also, let $m,M$ be such that for every $i$, $m \le |c_i| \le M$.
My intuition is that the bound may be related to the largest value of a polynomial approximating a constant function on some interval, but I am not sure. Can you find some nonzero $A$ (that may depend of course on $\epsilon,m,M,d$) such that $|p(a)| \geq A$?
Thank you very much.