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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C$ be an [[open set|02BH3MCBQB]] and let $f$ be [[holomorphic|0NAHXR1FRH]] on $\\Omega$, not identically zero on any connected component of $\\Omega$.\nThen the function $u = \\log\\abs{f}$, with $u(z) = -\\infty$ where $f(z) = 0$, is [[subharmonic|Z1MHMGFNFW]] on $\\Omega$.\nMoreover $u$ is [[harmonic|0DEB0JMN0W]] on the open set $\\set{z \\in \\Omega : f(z) \\ne 0}$.\n\n## Remarks\n\nProof idea: let $a$ be a point with $f(a) \\ne 0$.\nOn a small disc about $a$ there is a holomorphic $g$ with $f = e^{g}$, so $u = \\operatorname{Re} g$ there.\nThe [[real part of a holomorphic function is harmonic|031BG625Y3]], so $u$ is harmonic near $a$.\nAt a zero $a$ of $f$, $u(a) = -\\infty$, and $-\\infty$ is at most any mean of $u$, so the sub-mean inequality holds at $a$.\nThe function $u$ is continuous as a map into $[-\\infty, \\infty)$, so it is upper semicontinuous.\n\nExample: for $f(z) = z$, $u(z) = \\log\\abs{z}$ is harmonic off $0$ and equals $-\\infty$ at $0$.\nIts mean over the circle of radius $r$ about $0$ is $\\log r \u003e -\\infty = u(0)$.\nComposing with the convex increasing function $t \\mapsto e^{pt}$ shows that $\\abs{f}^p$ is subharmonic for every $p \u003e 0$, by [[convex functions of subharmonic functions|E820CTMJVM]].\nThe function $\\log\\abs{f}$ is not harmonic on all of $\\Omega$ when $f$ has zeros.\nIts Laplacian in the sense of distributions is $2\\pi \\sum_a m_a \\delta_a$, where $a$ runs over the zeros of $f$ and $m_a$ is the multiplicity of $a$.\n\n## Explanations\n\n### L1 human\n\nTake a holomorphic function, look at the size $\\abs{f}$ of its values, and take the logarithm.\nThe result is harmonic, except at the zeros of $f$, where it dives down to $-\\infty$.\nThose dives only push values below the average, so the function is subharmonic.\n\n### L2 human\n\nThis is the main bridge from complex analysis to potential theory.\nApplied to $\\log\\abs{f}$, the maximum principle for subharmonic functions gives the maximum modulus principle, and the convexity results for subharmonic functions give Hadamard's three-circle theorem.\nJensen's formula is the mean value formula for $\\log\\abs{f}$, with the zeros of $f$ as the correction term.\n",
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