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  "body": "\n## Summary\n\nLet $g$ be a univalent function on the outside of the unit disc.\nIts Grunsky matrix lists the coefficients of the double series of $\\log\\frac{g(z) - g(\\zeta)}{z - \\zeta}$.\nUnivalence is reflected in the Grunsky inequalities, which say that a weighted form of this matrix is a contraction on the sequence space $\\ell^2$.\nThese inequalities refine the [[Grönwall area theorem|BYZDAZNHP4]] and were a main tool in the work on coefficient problems before the proof of the [[Bieberbach conjecture|ZMRFSYVWQT]].\n\n## Explanations\n\n### L1 human\n\nA univalent function $g$ never takes the same value twice, so the difference quotient $(g(z) - g(\\zeta))/(z - \\zeta)$ never vanishes.\nIts logarithm is then a nice function of two variables, which can be expanded in powers of $1/z$ and $1/\\zeta$.\nThe Grunsky matrix lists the coefficients of this expansion.\nFor $g(z) = z + 1/z$ the expansion is $\\log(1 - 1/(z\\zeta))$, and the matrix is diagonal with entries $1, 1/2, 1/3, \\ldots$.\n",
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