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  "body": "\n## Definition\n\nLet $\\Omega$ be an [[open set|02BH3MCBQB]] in $\\C$.\nA function $u \\colon \\Omega \\to [-\\infty, \\infty)$ is *subharmonic* if:\n\n1. $u$ is upper semicontinuous, that is, $\\set{z \\in \\Omega : u(z) \u003c c}$ is open for every real $c$.\n2. $u$ satisfies the sub-mean value inequality: for every $a \\in \\Omega$ and $r \u003e 0$ with $\\set{z : \\abs{z - a} \\le r} \\subseteq \\Omega$,\n$$u(a) \\le \\frac{1}{2\\pi}\\int_0^{2\\pi} u(a + r e^{it})\\, dt.$$\n\n## Remarks\n\nSome authors also require that $u$ is not identically $-\\infty$ on any connected component of $\\Omega$.\nThe same definition, with averages over spheres, gives subharmonic functions on open sets of $\\R^n$.\nIt is equivalent to require the inequality only for all small enough $r$ at each point.\nExample: $u(z) = \\abs{z}^2$ is continuous, and its mean over the circle of radius $r$ about $a$ is $\\abs{a}^2 + r^2 \\ge \\abs{a}^2$, so $u$ is subharmonic.\nNon-example: $u(z) = -\\abs{z}^2$ is continuous, but its mean over the circle of radius $r$ about $0$ is $-r^2 \u003c 0 = u(0)$, so clause 2 fails.\nA function $u$ of class $C^2$ is subharmonic if and only if its [[Laplacian|0E0B6RA8K0]] satisfies $\\Delta u \\ge 0$.\nA function is [[harmonic|0DEB0JMN0W]] if and only if both $u$ and $-u$ are subharmonic and $u$ is finite.\nFor a holomorphic function $f$ on $\\Omega$, the function $\\log\\abs{f}$ is subharmonic, with value $-\\infty$ at the zeros of $f$.\n",
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