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  "body": "\n## Statement\n\nLet $0 \u003c r_1 \u003c r_2$ and let $u$ be [[subharmonic|Z1MHMGFNFW]] on an open subset of $\\C$ that contains the closed [[annulus|0CW52NV5TC]] $A = \\set{z : r_1 \\le \\abs{z} \\le r_2}$.\nFor $r_1 \\le r \\le r_2$ let $M(r) = \\max_{\\abs{z} = r} u(z)$, and assume that $M(r_1)$ and $M(r_2)$ are finite.\nThen for every $r$ with $r_1 \\le r \\le r_2$,\n$$M(r) \\le \\frac{\\log(r_2/r)}{\\log(r_2/r_1)}\\, M(r_1) + \\frac{\\log(r/r_1)}{\\log(r_2/r_1)}\\, M(r_2).$$\nEquivalently, $M(r)$ is a [[convex function|07Q7CNM5WF]] of $\\log r$ on $[r_1, r_2]$.\n\n## Remarks\n\nThe maximum $M(r)$ exists because an upper semicontinuous function attains its maximum on a compact circle.\nProof idea: choose real $a, b$ such that $h(z) = a + b \\log\\abs{z}$ equals $M(r_1)$ on $\\abs{z} = r_1$ and $M(r_2)$ on $\\abs{z} = r_2$.\nThe function $h$ is harmonic on $\\C \\setminus \\set{0}$, so $u - h$ is subharmonic on the open annulus, and its upper limits at the two boundary circles are at most $0$.\nThe [[maximum principle for subharmonic functions|9XJ5FBQD7M]] gives $u \\le h$ on $A$, so $M(r) \\le a + b \\log r$, which is the stated inequality.\nApplying the same bound to every sub-annulus gives convexity in $\\log r$.\n\nExample: for $u(z) = \\log\\abs{z}$, $M(r) = \\log r$ is linear in $\\log r$ and equality holds.\nFor a holomorphic $f$, the function $\\log\\abs{f}$ is [[subharmonic|7TXCCCA17A]], and the theorem becomes the classical [[Hadamard three-circle theorem|V00Q05ZW1G]] for $\\log \\max_{\\abs{z}=r} \\abs{f(z)}$.\n\n## Explanations\n\n### L1 human\n\nTake the largest value of a subharmonic function on each circle around the origin.\nPlotted against the logarithm of the radius, these largest values form a convex curve.\nSo the maximum on a middle circle is controlled by the maxima on an inner and an outer circle.\n\n### L2 human\n\nThe proof is a direct comparison with the harmonic functions $a + b\\log\\abs{z}$, which are the radial harmonic functions in the plane.\nThe same argument in $\\R^n$ with $n \\ge 3$ shows that the maximum over spheres is convex as a function of $r^{2-n}$.\nThe result is a standard tool for growth estimates of holomorphic and subharmonic functions, together with the Phragmén–Lindelöf principle.\n",
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