Fix $\epsilon >0$. Suppose $p_{n}(z) = z^{n}+a_{n-1}z^{n-1}+ \cdots + a_{0} \in \mathbb{Z}[x]$ is irreducible and has all positive real roots. Show that independently of $n$ except for finitely many explicitly computable exceptions that $|a_{n-1}| \geq (2-\epsilon)n$.
The sum of the roots is $-\frac{a_{n-1}}{a_n}$ which is $-a_{n-1}$. So $-a_{n-1} >0$ which means that $a_{n-1} \leq 0$. So $|a_{n-1}| = -a_{n-1}$. The problem can then be rephrased as follows: Show that the sum of the roots of the above polynomial is $ \geq (2-\epsilon)n$. Could we use a proof by contradiction? I think irreducibility is an important property here.
Source: Computational excursions in analysis and number theory by P Borwein