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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]], and let $u \\colon \\Omega \\to [-\\infty, \\infty)$ be a function.\nA *least harmonic majorant* of $u$ is a [[harmonic majorant|ZDHPQWWCQW]] $h_0$ of $u$ such that $h_0 \\le h$ on $\\Omega$ for every harmonic majorant $h$ of $u$.\n\n## Remarks\n\nA least harmonic majorant is unique when it exists, since two of them are each at most the other.\nIf $\\Omega$ is connected and $u$ is [[subharmonic|Z1MHMGFNFW]] with at least one harmonic majorant, then $u$ has a least harmonic majorant.\nThe *greatest harmonic minorant* of a function $v$ is defined in the same way; it is $-h_0$, where $h_0$ is the least harmonic majorant of $-v$.\n\nExample: on the unit disc $\\mathbb{D} \\subseteq \\C$, the least harmonic majorant of $u(z) = \\abs{z}^2$ is the constant $1$.\nIt is a harmonic majorant.\nLet $h$ be any harmonic majorant, and let $\\abs{a} \u003c r \u003c 1$.\nSince $h \\ge r^2$ on the circle of radius $r$, the [[Poisson integral formula|PCZBE7EX01]] on that circle gives $h(a) \\ge r^2$.\nLetting $r \\to 1$ gives $h(a) \\ge 1$.\n\nNon-example: the constant $2$ is a harmonic majorant of $\\abs{z}^2$ on $\\mathbb{D}$, but it is not least, because $1$ is also a harmonic majorant and $2 \\le 1$ fails.\n",
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