Determination of a neighborhood of the imaginary axis which is disjoint from the spectrum of a real polynomial

The distance of the spectrum of $f$ from the imaginary axis is estimated for a real polynomial $f(z)=\sum_{\nu=0}^na_\nu z^\nu$ with roots in the right (or as a corollary, in the left) half plane: $f:\min\operatorname{Resp}(f)\ge-1/\operatorname{tr}(H_1H^{-1})>0$ where $H:=[a_{n+i-2j}]_{i,j=\overline{1,n}}$ and $H_1:=[ka_k]$, $k:=n+i-2j+1$, $i,j=\overline{1,n}$.