= Hadamard three-circle theorem
{c}
{title2=$M(r)\leq M(r_0)^{1-\theta}M(r_1)^\theta,\quad\theta=\log(r/r_0)/\log(r_1/r_0)$}
For a holomorphic scalar function on a neighborhood of a closed annulus, $M(r)$ is the maximum modulus on its radius-$r$ circle. The logarithm of its modulus is <subharmonic>; compare it with the harmonic function affine in $\log|z|$ which takes the logarithms of the two boundary maxima. The <maximum principle for subharmonic functions> proves the displayed bound. Zero boundary maxima are handled by adding a positive bound and taking its limit. The same bound holds for the norm of a <Banach space>-valued holomorphic function by applying scalar functionals and the <Hahn-Banach theorem>.
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