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  "body": "\n## Statement\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$ be the [[unit disc|8G5ABXRK85]].\nThere is an absolute constant $b \u003e 0$ with the following property.\nLet $f \\colon \\mathbb{D} \\to \\C$ be any [[holomorphic|0DG0Z72CQD]] function with $f'(0) = 1$.\nThen there is an open set $\\Omega \\subseteq \\mathbb{D}$ such that $f$ is [[injective|0PVG1CRAJN]] on $\\Omega$ and $f(\\Omega)$ is an open disc of radius $b$.\n\n## Remarks\n\nThe supremum of all such $b$ is the Bloch constant $B$, one of [[Landau's constants|AC3FNJ8Y1Y]].\nLars Ahlfors proved $B \\ge \\sqrt{3}/4$, and it is known that $B \u003c 0.472$; the exact value is unknown.\nOne proof first replaces $f$ by $f(rz)/r$ for some $r \u003c 1$, so that $f$ is holomorphic on the closed disc.\nThen one picks a point where $(1 - \\abs{z}^2)\\abs{f'(z)}$ is maximal and rescales around it, which gives a function whose derivative is bounded on a disc of fixed size.\nA bounded derivative with $\\abs{f'(0)} = 1$ forces $f$ to be injective on a smaller disc of explicit radius, and the image of that disc contains a disc of explicit radius.\nThe theorem gives a short proof of [[Picard's little theorem|R2PHM118NG]].\nThe normalisation $f'(0) = 1$ is needed: the functions $f(z) = \\varepsilon z$ only cover discs of radius $\\varepsilon$.\n\n## Explanations\n\n### L1 human\n\nTake any holomorphic function on the unit disc whose derivative at the centre is $1$.\nIts image may fold over itself in a complicated way.\nBloch's theorem says that, somewhere inside the image, there is always a round disc of a fixed size that is covered exactly once.\nThe size of this disc does not depend on the function.\n\n### L2 human\n\nThe theorem is a covering theorem with no assumption of injectivity on $f$.\nFor injective $f$, the [[Koebe quarter theorem|WX0HMW4GT7]] gives much more: a disc of radius $1/4$ around $f(0)$.\nBloch's theorem is scale invariant in the right way, so applying it to $f(rz)$ with large $r$ shows that a nonconstant entire function covers schlicht discs of every radius.\nThis is the key step in the proof of Picard's little theorem by Bloch's method.\n",
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