Area theorem (conformal mapping) (${\sum }_{n\ge 1}n|b_n{|}^2\le 1$)

For a univalent exterior map $G(\zeta )=\zeta +b_0+{\sum }_{n\ge 1}{b}_n{\zeta }^{-n}$, the image of $|\zeta |=R>1$ is a positively oriented Jordan curve. Green's theorem gives its enclosed area as $\pi (R^2-{\sum }_{n\ge 1}n|b_n{|}^2{R}^{-2n})$. Nonnegativity and $R\downarrow 1$ give the displayed bound. Laurent-series differentiation and integration on a circle of radius greater than one are justified by uniform convergence there.