Hadamard three-circle theorem

In complex analysis, a branch of mathematics, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions.

Let ${\displaystyle f(z)}$ be a holomorphic function on the annulus

${\displaystyle r_{1}\leq \left|z\right|\leq r_{3}.}$

Let ${\displaystyle M(r)}$ be the maximum of ${\displaystyle |f(z)|}$ on the circle ${\displaystyle |z|=r.}$ Then, ${\displaystyle \log M(r)}$ is a convex function of the logarithm ${\displaystyle \log(r).}$ Moreover, if ${\displaystyle f(z)}$ is not of the form ${\displaystyle cz^{\lambda }}$ for some constants ${\displaystyle \lambda }$ and ${\displaystyle c}$, then ${\displaystyle \log M(r)}$ is strictly convex as a function of ${\displaystyle \log(r).}$

The conclusion of the theorem can be restated as

${\displaystyle \log \left({\frac {r_{3}}{r_{1}}}\right)\log M(r_{2})\leq \log \left({\frac {r_{3}}{r_{2}}}\right)\log M(r_{1})+\log \left({\frac {r_{2}}{r_{1}}}\right)\log M(r_{3})}$

for any three concentric circles of radii ${\displaystyle r_{1}<r_{2}<r_{3}.}$