= Area theorem (conformal mapping)
{title2=$\sum_{n\ge1}n|b_n|^2\le1$}
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= Area theorem for univalent functions
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= Grönwall area theorem
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{synonym}
For a <univalent> exterior map $G(\zeta)=\zeta+b_0+\sum_{n\ge1}b_n\zeta^{-n}$, the image of $|\zeta|=R>1$ is a positively oriented Jordan curve. <Green's theorem> gives its enclosed area as $\pi(R^2-\sum_{n\ge1}n|b_n|^2R^{-2n})$. Nonnegativity and $R\downarrow1$ give the displayed bound. Laurent-series differentiation and integration on a <circle> of radius greater than one are justified by <uniform convergence> there.
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