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  "body": "\n## Statement\n\nLet $\\Omega \\subsetneq \\C$ be a [[simply connected domain|0QW64ARSHB]], and let $\\varphi \\colon \\Omega \\to \\mathbb{D}$ be a [[conformal map|0YYT36R4YD]] onto the unit disc $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$.\nThen for every real $p$ with $4/3 \u003c p \u003c 4$,\n$$\\int_\\Omega \\abs{\\varphi'(z)}^p \\, dA(z) \u003c \\infty,$$\nwhere $dA$ is area measure.\n\n## Remarks\n\nSuch a map $\\varphi$ exists by the [[Riemann mapping theorem|02WHMEG5ZR]].\nFor $p = 2$ the integral is the area of $\\mathbb{D}$, which is $\\pi$.\nThe range is sharp at both ends.\nTake the Koebe domain $\\Omega = \\C \\setminus (-\\infty, -1/4]$ and the inverse $\\varphi$ of the [[Koebe function|EZQFWKF03Q]].\nThen $\\abs{\\varphi'(z)}$ behaves like $\\abs{z}^{-3/2}$ at infinity and like $\\abs{z + 1/4}^{-1/2}$ near the tip of the slit.\nSo the integral diverges for $p \\le 4/3$ and for $p \\ge 4$.\nThe conjecture is known for $4/3 \u003c p \u003c 3 + \\delta$ with some $\\delta \u003e 0$; Brennan proved this range.\nThe full conjecture is open.\n\n## Explanations\n\n### L1 human\n\nThe derivative of a Riemann map measures how much the map stretches the domain near each point.\nNear a sharp inward corner or the tip of a slit, this stretching blows up.\nThe conjecture says that it never blows up too badly: a certain power of it always has finite total integral.\nThe slit plane is the worst case and shows that the claimed range of powers cannot be enlarged.\n",
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      "first_stated": "1978",
      "posed_by": "James E. Brennan"
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    "msc": [
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    "name": "Brennan conjecture",
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      "wikipedia": "Brennan conjecture"
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  "name": "Brennan conjecture",
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