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  "body": "\n## Statement\n\nLet $\\alpha \u003e \\frac{1}{2}$ and let $S = \\set{z \\in \\C \\setminus \\set{0} : \\abs{\\arg z} \u003c \\frac{\\pi}{2\\alpha}}$ be the open sector of opening $\\pi / \\alpha$.\nLet $f$ be continuous on the closure $\\bar S$ and [[holomorphic|0DG0Z72CQD]] on $S$.\nAssume that $\\abs{f(z)} \\le M$ for each $z$ on the boundary of $S$.\nAssume also that there are constants $C \u003e 0$ and $\\beta \u003c \\alpha$ with $\\abs{f(z)} \\le C \\exp(\\abs{z}^{\\beta})$ for all $z \\in S$.\nThen $\\abs{f(z)} \\le M$ for all $z \\in S$.\n\n## Remarks\n\nProof idea: pick $\\gamma$ with $\\beta \u003c \\gamma \u003c \\alpha$ and put $g_\\varepsilon(z) = f(z) \\exp(-\\varepsilon z^{\\gamma})$ for $\\varepsilon \u003e 0$.\nOn $\\bar S$ one has $\\operatorname{Re} z^{\\gamma} \\ge \\cos\\left(\\frac{\\gamma \\pi}{2\\alpha}\\right) \\abs{z}^{\\gamma}$ with a positive cosine, so $g_\\varepsilon \\to 0$ at infinity.\nThe [[maximum modulus principle|048F2P90WR]] on bounded pieces of $S$ gives $\\abs{g_\\varepsilon} \\le M$, and $\\varepsilon \\to 0$ gives the claim.\nThe growth condition cannot be relaxed to $\\beta = \\alpha$.\nThe function $\\exp(z^{\\alpha})$ has modulus $\\exp(r^{\\alpha} \\cos(\\alpha \\theta))$ at $z = r e^{i\\theta}$, which is $1$ on the boundary rays and unbounded on the positive axis.\nThere are similar statements for strips, half-planes and other unbounded domains; the [[Hadamard three-lines theorem|WHS58R43N0]] is one of them.\n\n## Explanations\n\n### L1 human\n\nOn a bounded region, the maximum principle says that a holomorphic function is bounded inside by its bound on the boundary.\nOn an unbounded region such as a sector, this can fail, because the function can grow fast toward infinity.\nThe Phragmén–Lindelöf principle says that the maximum principle still holds if the function does not grow too fast.\nThe allowed speed depends on the opening of the sector: the narrower the sector, the faster the allowed growth.\n\n### L2 human\n\nThe principle is used throughout the theory of entire functions and in analytic number theory.\nIt gives convexity bounds for $L$-functions in vertical strips, and it underlies results such as [[Lindelöf's theorem|XBM7J5R1VQ]] and Carlson's theorem.\n",
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