G.S. Carr's "Synopsis of Elementary Results in Pure Mathematics" mentions the following bound on the roots of a polynomial in its article 449:
If $x^{n-r}$ and $x^{n-s}$ are the highest negative terms in $f(x)$ and $f(-x)$ respectively, $(1+ \sqrt[r] p)$ and $-(1+ \sqrt[s] q)$ are limits of the roots.
Of course ,$f(x) := x^n +p_1x^{n-1}+...+p_n$, and $p,q$ are the coefficients of $x^{n-r}, x^{n-s}$ respectively, i.e. $|p_r|, |p_s|$ in truth.
This bound resembles both the Cauchy and the Lagrange-Zassenhauss bounds. But I couldn't find it in a textbook or on the net. Any ideas how to prove it? Thanks!