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  "body": "\n## Definition\n\nLet $\\Delta = \\set{z \\in \\C : \\abs{z} \u003e 1}$.\nLet $g$ be an injective meromorphic function on $\\Delta \\cup \\set{\\infty}$ with a simple pole at $\\infty$ and [[Laurent expansion|0X3XD2ATB0]]\n$$g(z) = z + b_0 + \\sum_{n=1}^{\\infty} b_n z^{-n}, \\qquad \\abs{z} \u003e 1.$$\nThe function $Q(z, \\zeta) = \\dfrac{g(z) - g(\\zeta)}{z - \\zeta}$ for $z \\ne \\zeta$, with $Q(z, z) = g'(z)$, is holomorphic and nowhere zero on $\\Delta \\times \\Delta$ and tends to $1$ as $z, \\zeta \\to \\infty$.\nSo it has a holomorphic [[logarithm|0X2FY491PF]] with value $0$ at $(\\infty, \\infty)$, with an expansion\n$$\\log \\frac{g(z) - g(\\zeta)}{z - \\zeta} = -\\sum_{m, n \\ge 1} c_{mn}\\, z^{-m} \\zeta^{-n}, \\qquad \\abs{z}, \\abs{\\zeta} \u003e 1.$$\nThe numbers $c_{mn}$ are the *Grunsky coefficients* of $g$, and the infinite matrix $(c_{mn})_{m, n \\ge 1}$ is the *Grunsky matrix* of $g$.\n\n## Remarks\n\nThe Grunsky matrix is symmetric: $c_{mn} = c_{nm}$, because $Q(z, \\zeta) = Q(\\zeta, z)$.\nThe coefficients are polynomials in $b_1, b_2, \\ldots$; for example $c_{11} = b_1$.\nThe Grunsky inequalities say that for every finite sequence $\\lambda_1, \\ldots, \\lambda_N \\in \\C$,\n$$\\Bigl| \\sum_{m, n = 1}^{N} \\sqrt{mn}\\, c_{mn} \\lambda_m \\lambda_n \\Bigr| \\le \\sum_{n=1}^{N} \\abs{\\lambda_n}^2.$$\nExample: $g(z) = z + 1/z$ is injective on $\\Delta$, since $g(z) = g(w)$ with $z \\ne w$ forces $zw = 1$.\nHere $Q(z, \\zeta) = 1 - 1/(z\\zeta)$, so $\\log Q = -\\sum_{n \\ge 1} \\frac{1}{n} (z\\zeta)^{-n}$, and $c_{nn} = 1/n$ while $c_{mn} = 0$ for $m \\ne n$.\nThe weighted matrix $(\\sqrt{mn}\\, c_{mn})$ is the identity, so the Grunsky inequalities reduce to $\\abs{\\sum_n \\lambda_n^2} \\le \\sum_n \\abs{\\lambda_n}^2$, with equality for real $\\lambda_n$.\nNon-example: $g(z) = z + 2/z$ is not injective on $\\Delta$, since $g(i\\sqrt{2}) = g(-i\\sqrt{2}) = 0$.\nThen $Q(z, \\zeta) = 1 - 2/(z\\zeta)$ vanishes at points of $\\Delta \\times \\Delta$, and no holomorphic logarithm exists there.\nSign and normalisation conventions differ between authors; some use $\\log\\frac{g(z) - g(\\zeta)}{z - \\zeta}$ with a plus sign or with the factors $\\sqrt{mn}$ built in.\n",
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