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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, usually [[subharmonic|Z1MHMGFNFW]].\nA *harmonic majorant* of $u$ is a [[harmonic function|0DEB0JMN0W]] $h$ on $\\Omega$ such that $u(x) \\le h(x)$ for every $x \\in \\Omega$.\n\n## Remarks\n\nSymmetrically, a harmonic minorant of $v$ is a harmonic $h$ with $h \\le v$, so $h$ is a harmonic minorant of $v$ exactly when $-h$ is a harmonic majorant of $-v$.\nFor a holomorphic $f$ on the unit disc and $0 \u003c p \u003c \\infty$, $f$ lies in the [[Hardy space|PNNNFJVF6W]] $H^p$ if and only if the subharmonic function $\\abs{f}^p$ has a harmonic majorant.\n\nExample: on the unit disc $\\mathbb{D} \\subseteq \\C$, let $u(z) = \\abs{z}^2$.\nThe constant $h = 1$ is harmonic and $\\abs{z}^2 \u003c 1$ on $\\mathbb{D}$, so $h$ is a harmonic majorant of $u$.\n\nNon-example: on all of $\\C$, $u(z) = \\abs{z}^2$ has no harmonic majorant.\nA harmonic $h$ with $h \\ge \\abs{z}^2 \\ge 0$ would be a harmonic function bounded below on $\\C$, hence constant by [[Liouville's theorem for harmonic functions|QK1KWYZNRD]].\nNo constant is at least $\\abs{z}^2$ for all $z$.\n",
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