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The sharp constant in the Chang--Marshall inequality
We prove that the optimal constant in the Chang--Marshall inequality is $e$, as conjectured by Andreev and Matheson. More precisely, if $f$ is analytic in the unit disk, vanishes to order $m\geq1$ at the origin, and has normalized Dirichlet integral $d\leq1$, then the boundary mean of $e^{|f|^2}$ is
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