Hadamard three-circle theorem
# 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 $f(z)$ be a holomorphic function on the annulus

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

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

The conclusion of the theorem can be restated as

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

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

## History

A statement and proof for the theorem was given by J.E. Littlewood in 1912, but he attributes it to no one in particular, stating it as a known theorem. Harald Bohr and Edmund Landau attribute the theorem to Jacques Hadamard, writing in 1896; Hadamard published no proof.

## Proof

The three circles theorem follows from the fact that for any real a, the function Re $log(z^af(z))$ is harmonic between two circles, and therefore takes its maximum value on one of the circles. The theorem follows by choosing the constant a so that this harmonic function has the same maximum value on both circles.

The theorem can also be deduced directly from Hadamard's three-lines theorem.

