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  "body": "\n## Statement\n\nLet $R \u003e 0$ and let $f$ be [[holomorphic|0DG0Z72CQD]] on an open set containing the closed disc $\\set{z : \\abs{z} \\le R}$.\nLet $A(R) = \\max_{\\abs{z} = R} \\operatorname{Re} f(z)$.\nThen for each $r$ with $0 \u003c r \u003c R$,\n$$\\max_{\\abs{z} = r} \\abs{f(z)} \\le \\frac{2r}{R - r} A(R) + \\frac{R + r}{R - r} \\abs{f(0)}.$$\n\n## Remarks\n\nProof idea: first let $f(0) = 0$ and $f$ not constant, so that $A(R) \u003e 0$ by the mean value property of $\\operatorname{Re} f$.\nThen $g = f / (2A(R) - f)$ maps the disc of radius $R$ into the unit disc with $g(0) = 0$.\nThe [[Schwarz lemma|0MB48XDBY3]] gives $\\abs{g(z)} \\le \\abs{z}/R$, and solving for $f$ gives $\\abs{f(z)} \\le \\frac{2r A(R)}{R - r}$ on $\\abs{z} = r$.\nIn general, apply this to $f - f(0)$, whose real part is at most $A(R) + \\abs{f(0)}$ on $\\abs{z} = R$.\nExample: a real constant $f = c \\le 0$ gives equality.\nThe left side is $\\abs{c}$, and the right side is $\\frac{2rc + (R + r)\\abs{c}}{R - r} = \\abs{c}$.\n\n## Explanations\n\n### L1 human\n\nUsually one needs a bound on $\\abs{f}$ to control $\\abs{f}$.\nThis theorem says that an upper bound on the real part alone is enough, on a slightly smaller disc.\nThe price is a factor that blows up as the smaller radius approaches the larger one.\n\n### L2 human\n\nThe theorem is a standard tool for entire functions and in analytic number theory.\nA typical use is to bound $\\log f$ when $f$ has no zeros in a disc.\nThere $\\operatorname{Re} \\log f = \\log \\abs{f}$ is easy to bound from above.\nThe theorem then bounds $\\abs{\\log f}$ and, through Cauchy's estimates, its derivatives.\nThis is a step in the proof of the [[Hadamard factorization theorem|KYSR7NHNFQ]] and in zero-free region arguments for the Riemann zeta function.\n",
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