Vol. 6, No. 4, 2013

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An elementary inequality about the Mahler measure

Konstantin Stulov and Rongwei Yang

Vol. 6 (2013), No. 4, 393–397
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 $td\left(p\right)={\sum }_{j=1}^{n}||{z}_{j}|-1|$. 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.

Keywords
Mahler measure, total distance
Primary: 11CXX