An elementary inequality about the Mahler measure

Abstract

Let $p\left(z\right)$ be a degree $n$ polynomial with zeros $z_j,j=1,2,\dots ,n$. The total distance from the zeros of $p$ to the unit circle is defined as $td\left(p\right)={\sum }_{j=1}^n\left|\left|z_j\right|-1\right|$. We show that up to scalar multiples, $td\left(p\right)$ sits between $M\left(p\right)-1$ and $m\left(p\right)$. This leads to an equivalent statement of Lehmer’s problem in terms of $td\left(p\right)$. The proof is elementary.