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  "body": "\n## Statement\n\nLet $x_1 \u003c x_2$ and $y_0$ be real numbers, and let $\\Omega = \\set{z \\in \\C : x_1 \\le \\operatorname{Re} z \\le x_2, \\ \\operatorname{Im} z \\ge y_0}$ be a closed half-strip.\nLet $f \\colon \\Omega \\to \\C$ be continuous on $\\Omega$ and [[holomorphic|0DG0Z72CQD]] on its interior.\nAssume that $\\abs{f(z)} \\le M$ for each $z$ on the boundary of $\\Omega$.\nAssume also that there are constants $A, B \u003e 0$ with $\\abs{f(x + iy)} \\le B (1 + \\abs{y})^{A}$ for all $x + iy \\in \\Omega$.\nThen $\\abs{f(z)} \\le M$ for all $z \\in \\Omega$.\n\n## Remarks\n\nProof idea: for $\\varepsilon \u003e 0$ let $g_\\varepsilon(z) = f(z) e^{i \\varepsilon (z - i y_0)}$, so that $\\abs{g_\\varepsilon(x + iy)} = \\abs{f(x + iy)} e^{-\\varepsilon (y - y_0)}$.\nThen $\\abs{g_\\varepsilon} \\le \\abs{f}$ on $\\Omega$, and $g_\\varepsilon \\to 0$ as $\\operatorname{Im} z \\to \\infty$ because $f$ grows at most polynomially.\nThe [[maximum modulus principle|048F2P90WR]] on the rectangles $\\Omega \\cap \\set{\\operatorname{Im} z \\le T}$ with $T$ large gives $\\abs{g_\\varepsilon} \\le M$, and $\\varepsilon \\to 0$ gives $\\abs{f} \\le M$.\nA growth hypothesis is needed.\nOn the half-strip $\\abs{\\operatorname{Re} z} \\le \\frac{\\pi}{2}$, $\\operatorname{Im} z \\ge 0$, the function $\\exp(e^{-iz})$ has modulus $\\exp(e^{y} \\cos x)$.\nIt is bounded on the boundary, where $\\cos x = 0$ or $y = 0$, but unbounded on the line $x = 0$.\nThe theorem is a case of the [[Phragmén–Lindelöf principle|FYR6V2NH6C]].\n\n## Explanations\n\n### L1 human\n\nOn a half-strip, which goes off to infinity in one direction, the maximum principle can fail for fast-growing functions.\nLindelöf's theorem says that polynomial growth is slow enough.\nThen a bound on the edges of the half-strip is a bound everywhere inside.\n\n### L2 human\n\nThe theorem is used in analytic number theory for functions such as the Riemann zeta function on vertical half-strips.\nTogether with the [[Hadamard three-lines theorem|WHS58R43N0]] it gives the convexity bounds for the growth of $\\zeta(\\sigma + it)$ as $t \\to \\infty$.\nIn the proof, the factor $e^{i\\varepsilon z}$ is the standard device to force decay at infinity before applying the maximum modulus principle.\n",
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