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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.doi:10.1215/ijm/1391178550 fatcat:xwrm6stonra5bprqba4stwuz3e