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  "body": "\n## Statement\n\nLet $g$ be an injective meromorphic function on $\\set{z \\in \\C : \\abs{z} \u003e 1}$ with [[Laurent expansion|0X3XD2ATB0]]\n$$g(z) = z + b_0 + \\sum_{n=1}^{\\infty} b_n z^{-n}, \\qquad \\abs{z} \u003e 1.$$\nThen\n$$\\sum_{n=1}^{\\infty} n \\abs{b_n}^2 \\le 1.$$\n\n## Remarks\n\nFor $r \u003e 1$, the image of the circle $\\abs{z} = r$ is a Jordan curve, and the bounded region $E_r$ it encloses is the complement of $g(\\set{\\abs{z} \u003e r})$.\nGreen's formula gives the area of $E_r$ as $\\frac{1}{2i}\\oint \\bar{w}\\, dw$ over this curve, which works out to $\\pi\\bigl(r^2 - \\sum_{n \\ge 1} n \\abs{b_n}^2 r^{-2n}\\bigr)$.\nThis area is nonnegative, and letting $r \\to 1$ gives the inequality.\nEquality holds exactly when the complement of $g(\\set{\\abs{z} \u003e 1})$ has area $0$.\nExample: $g(z) = z + 1/z$ maps $\\set{\\abs{z} \u003e 1}$ onto $\\C \\setminus [-2, 2]$, and $b_1 = 1$ gives equality.\nIn particular $\\abs{b_1} \\le 1$.\nApplied to $g(z) = 1/\\sqrt{f(1/z^2)}$ for $f$ in the class $S$ of normalised [[univalent functions|R91E83SDBD]], it gives Bieberbach's inequality $\\abs{a_2} \\le 2$.\n\n## Explanations\n\n### L1 human\n\nThe function $g$ maps the outside of the unit disc onto the outside of some compact set in the plane.\nThat compact set has a nonnegative area, and the area can be written with the coefficients of $g$.\nSo these coefficients cannot be too large, which is the inequality.\n\n### L2 human\n\nThe area theorem is the starting point of the coefficient theory of univalent functions.\nThrough the substitution $f \\mapsto 1/\\sqrt{f(1/z^2)}$, it gives $\\abs{a_2} \\le 2$ for $f \\in S$, hence the [[Koebe quarter theorem|WX0HMW4GT7]] and the distortion theorems.\nThe Grunsky inequalities on the [[Grunsky matrix|XKRR37N916]] refine it to a statement about all the coefficients of $\\log\\frac{g(z) - g(\\zeta)}{z - \\zeta}$.\nThe equality case shows that the bound $\\abs{b_1} \\le 1$ is attained exactly by maps onto the complement of a segment.\n",
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