{
  "aliases": [
    "cauchy-inequalities",
    "cauchy-estimate"
  ],
  "backlinks": [
    {
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.\nLet $c \\in \\mathbb{C}$ and $r \u003e 0$ be such that the closed disc $\\{z : |z - c| \\le r\\}$ is contained in $U$.\nLet $M \\ge 0$ be a real number with $|f(z)| \\le M$ for each $z$ with $|z - c| = r$.\nThen for each integer $n \\ge 0$,\n$$|f^{(n)}(c)| \\le \\frac{n! \\, M}{r^n}.$$\n\n## Remarks\n\nProof: apply the [[estimation lemma|0DJMGPN8K9]] to the [[formula for the derivatives|0F0H5P5DED]] with $a = c$. The circle has length $2 \\pi r$, and $|f(z) / (z - c)^{n+1}| \\le M / r^{n+1}$ on it.\nFor $n = 1$, the estimate is $|f'(c)| \\le M / r$.\nIf $f(z) = \\sum a_n (z - c)^n$ is the Taylor series, the estimate is $|a_n| \\le M / r^n$.\n\n## Explanations\n\n### L1 human\n\nIf a holomorphic function is small on a circle, then its derivatives at the center cannot be large.\nA larger circle with the same bound $M$ gives a better bound for the derivatives. This is the idea of [[Liouville's theorem|0EHKVRF4J6]].\n",
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    "aliases": [
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    "name": "Cauchy's estimates",
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        "locator": "Consequences",
        "url": "https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula"
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  "markdown": "https://math2.tamnd.com/node/0JJRENWMP6.md",
  "msc": [
    "30A10"
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  "name": "Cauchy's estimates",
  "prov": null,
  "role": "corollary",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JJ/0JJRENWMP6.md",
  "status": null,
  "tag": "0JJRENWMP6",
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  "url": "https://math2.tamnd.com/wiki/cauchys-estimates/",
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