Two inequalities on the areal Mahler measure

TL;DR

This paper investigates the areal Mahler measure, especially for small values, providing improved inequalities that could impact understanding related to Lehmer's conjecture.

Contribution

It offers refined bounds on the areal Mahler measure, advancing the theoretical understanding of this function in the context of Lehmer's conjecture.

Findings

Improved inequalities for the areal Mahler measure when its value is small

Enhanced bounds that could influence Lehmer's conjecture

Deeper analysis of the areal Mahler measure's properties

Abstract

Recent work of Pritsker defines and studies an areal version of the Mahler measure. We further explore this function with a particular focus on the case where its value is small, as this is most relevant to Lehmer's conjecture. In this situation, we provide improvements to two inequalities established in Pritsker's original paper.