## Involve: A Journal of Mathematics

• Involve
• Volume 6, Number 4 (2013), 393-397.

### An elementary inequality about the Mahler measure

#### Abstract

Let $p(z)$ be a degree $n$ polynomial with zeros $zj,j=1,2,…,n$. The total distance from the zeros of $p$ to the unit circle is defined as $td(p)= ∑j=1n||zj|−1|$. We show that up to scalar multiples, $td(p)$ sits between $M(p)−1$ and $m(p)$. This leads to an equivalent statement of Lehmer’s problem in terms of $td(p)$. The proof is elementary.

#### Article information

Source
Involve, Volume 6, Number 4 (2013), 393-397.

Dates
Revised: 12 February 2013
Accepted: 16 February 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513733607

Digital Object Identifier
doi:10.2140/involve.2013.6.393

Mathematical Reviews number (MathSciNet)
MR3115974

Zentralblatt MATH identifier
1327.11077

Subjects
Primary: 11CXX

Keywords
Mahler measure total distance

#### Citation

Stulov, Konstantin; Yang, Rongwei. An elementary inequality about the Mahler measure. Involve 6 (2013), no. 4, 393--397. doi:10.2140/involve.2013.6.393. https://projecteuclid.org/euclid.involve/1513733607

#### References

• G. Everest and T. Ward, Heights of polynomials and entropy in algebraic dynamics, Springer, London, 1999.
• J. B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics 236, Springer, New York, 2007.
• C. Smyth, “The Mahler measure of algebraic numbers: a survey”, pp. 322–349 in Number theory and polynomials, London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008.