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  "body": "\n## Statement\n\nLet $\\Omega \\subset \\C$ be a bounded [[domain|0Q4CBD3HX0]] and let $F \\subset \\partial\\Omega$ be a finite set.\nLet $u$ be a [[subharmonic function|Z1MHMGFNFW]] on $\\Omega$ that is bounded above.\nAssume that $\\limsup_{z \\to \\zeta} u(z) \\le 0$ for every $\\zeta \\in \\partial\\Omega \\setminus F$.\nThen $u \\le 0$ on $\\Omega$.\n\n## Remarks\n\nProof idea: let $d$ be the diameter of $\\Omega$ and let $\\varepsilon \u003e 0$.\nThe function $v = u + \\varepsilon \\sum_{p \\in F} \\log(\\abs{z - p}/d)$ is subharmonic, and $v \\le u$ on $\\Omega$ because each logarithm is negative there.\nAt each $p \\in F$, $v \\to -\\infty$ since $u$ is bounded above; so $\\limsup v \\le 0$ at every boundary point.\nThe [[maximum principle for subharmonic functions|9XJ5FBQD7M]] gives $v \\le 0$, and letting $\\varepsilon \\to 0$ gives $u \\le 0$.\nThe bound from above is needed.\nOn the punctured disc $\\Omega = \\set{0 \u003c \\abs{z} \u003c 1}$ with $F = \\set{0}$, the function $u(z) = -\\log\\abs{z}$ is harmonic, tends to $0$ on the unit circle, and is positive on $\\Omega$.\nThe finite set $F$ may be replaced by any [[polar set|C95QJ7YH5T]] in $\\partial\\Omega$, with the same conclusion.\n\n## Explanations\n\n### L1 human\n\nThe ordinary maximum principle needs control at every boundary point.\nThis version says that finitely many bad boundary points do not matter, provided the function cannot blow up there.\n\n### L2 human\n\nThe lemma is the bounded form of the [[Phragmén–Lindelöf principle|FYR6V2NH6C]]: a growth condition lets one ignore a small part of the boundary.\nApplied to $\\log\\abs{f}$ for a bounded holomorphic $f$, it shows that $\\abs{f} \\le 1$ on $\\Omega$ when $\\limsup \\abs{f} \\le 1$ at all but finitely many boundary points.\nA related result for holomorphic functions on a half-strip is [[Lindelöf's theorem|XBM7J5R1VQ]].\nIn the study of harmonic measure the lemma shows that finite sets of boundary points carry no mass.\n",
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