Abstract: We consider various generalizations of the classical Mahler measure,
and prove sharp lower bounds for them. For example, we show that for any monic polynomial $P \in {\mathbb C}[z]$ satisfying $|P(0)|=1$ the quantity
$$M_0(P) = \exp\left(\frac{1}{2\pi}\int_0^{2\pi} \log^+|P(e^{i\theta})|\,d\theta\right)$$
is greater than or equal to $M(1+z_1+z_2)=1.381356\dots$. This inequality is best possible, and equality is attained for all $P(z)=z^n+c$ with $n \in {\mathbb N}$ and $|c|=1$.
(Joint work with Arturas Dubickas.)
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