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An explicit formula for the Mahler measure of the $3$-variable Laurent polynomial $a+b{x}^{-1}+cy+(a+bx+cy)z$ is given, in terms of dilogarithms and trilogarithms.

Let $M\left(\alpha \right)$ be the Mahler measure of an algebraic number $\alpha$, and $V$ be an open subset of $ℂ$. Then its $L\left(V\right)$ is inf $M{\left(\alpha \right)}^{1/deg\left(\alpha \right)}$, the infimum being over all non-zero non-cyclotomic $\alpha$ lying with its conjugates outside $V$. We evaluate $L\left(V\right)$ when $V$ is any annulus centered at $0$. We do the same for a variant of $L\left(V\right)$, which we call the transfinite Lehmer constant $L_{\infty }\left(V\right)$.Also, we prove the converse to Langevin’s Theorem, which states that $L\left(V\right)\>1$ if $V$ contains a point of modulus $1$. We prove the corresponding result for $L_{\infty }\left(V\right)$.