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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]].\n\n1. Let $u$ be [[subharmonic|Z1MHMGFNFW]] on $\\Omega$ and let $\\varphi \\colon \\R \\to \\R$ be [[convex|07Q7CNM5WF]] and increasing.\nPut $\\varphi(-\\infty) = \\lim_{t \\to -\\infty} \\varphi(t) \\in [-\\infty, \\infty)$.\nThen $\\varphi \\circ u$ is subharmonic on $\\Omega$.\n2. Let $h$ be [[harmonic|0DEB0JMN0W]] on $\\Omega$ and let $\\varphi \\colon \\R \\to \\R$ be convex.\nThen $\\varphi \\circ h$ is subharmonic on $\\Omega$.\n\n## Remarks\n\nProof idea: a convex function on $\\R$ is continuous, and an increasing continuous function preserves upper semicontinuity.\nFor the sub-mean inequality on a sphere with normalised measure $\\sigma$, monotonicity and [[Jensen's inequality|0RWMRTPC87]] give\n$$\\varphi(u(a)) \\le \\varphi\\Big(\\int u \\, d\\sigma\\Big) \\le \\int \\varphi(u) \\, d\\sigma.$$\nIn part 2 the first step is an equality, by the mean value property of $h$, so monotonicity of $\\varphi$ is not needed.\nIf $\\int u \\, d\\sigma = -\\infty$, then $u(a) = -\\infty$ and $\\varphi(u(a)) = \\inf \\varphi$, which is below every value of $\\varphi \\circ u$.\n\nExamples: if $u$ is subharmonic, so are $e^{u}$ and $\\max(u, 0)$; if $h$ is harmonic, so are $\\abs{h}$ and $h^2$.\nMonotonicity is needed in part 1: $u(x) = \\abs{x}^2$ is subharmonic and $\\varphi(t) = -t$ is convex, but $\\varphi \\circ u = -\\abs{x}^2$ is not subharmonic.\n\n## Explanations\n\n### L1 human\n\nSubharmonic functions sit below their averages.\nApplying an increasing convex function keeps them below their averages, because a convex function of an average is at most the average of the convex function.\nSo for example $e^u$ is subharmonic whenever $u$ is.\n\n### L2 human\n\nThis is the multi-dimensional analogue of the fact that an increasing convex function of a convex function is convex.\nCombined with the subharmonicity of $\\log\\abs{f}$ for holomorphic $f$, it shows that $\\abs{f}^p$ is subharmonic for all $p \u003e 0$, which is the starting point of the theory of Hardy spaces.\nIt also shows that $\\log u$ need not be subharmonic when $u$ is, since $\\log$ is concave; functions with $\\log u$ subharmonic form the smaller class of log-subharmonic functions.\n",
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