---
tag: V00Q05ZW1G
kind: statement
role: theorem
name: Hadamard three-circle theorem
msc: [30C80, 30A10]
uses: [0DG0Z72CQD, 048F2P90WR, 07Q7CNM5WF]
about: [0NAHXR1FRH]
sources:
  - url: https://en.wikipedia.org/wiki/Hadamard_three-circle_theorem
same_as:
  wikidata: Q729359
  wikipedia: Hadamard three-circle theorem
---

## Statement

Let $0 < r_1 < r_2$ and let $f$ be [[holomorphic|0DG0Z72CQD]] on an open set containing the closed annulus $\set{z : r_1 \le \abs{z} \le r_2}$.
For $r_1 \le r \le r_2$ let $M(r) = \max_{\abs{z} = r} \abs{f(z)}$.
Then for each $r$ with $r_1 \le r \le r_2$,
$$\log M(r) \le \frac{\log(r_2 / r)}{\log(r_2 / r_1)} \log M(r_1) + \frac{\log(r / r_1)}{\log(r_2 / r_1)} \log M(r_2).$$
Equivalently, $\log M(r)$ is a [[convex function|07Q7CNM5WF]] of $\log r$.

## Remarks

Proof idea: choose a real $\alpha$ such that $r_1^{\alpha} M(r_1) = r_2^{\alpha} M(r_2)$.
For rational $\alpha = p/q$ with $q > 0$, the [[maximum modulus principle|048F2P90WR]] for $z^p f(z)^q$ on the annulus gives $r^{\alpha} M(r) \le r_1^{\alpha} M(r_1)$.
A limit over rational $\alpha$, or the subharmonicity of $\log \abs{f(z)} + \alpha \log \abs{z}$, handles irrational $\alpha$, and the bound rearranges to the inequality.
Example: for $f(z) = z^n$, $\log M(r) = n \log r$ is linear in $\log r$, so equality holds.
If $M(r) = 0$ for some $r$, then $f$ vanishes on a circle, so $f \equiv 0$ on the annulus and the inequality is trivial.

## Explanations

### L1 human

Take three circles around $0$, one inside another.
The theorem bounds how large $\abs{f}$ can be on the middle circle by its sizes on the inner and outer circles.
Plot the logarithm of the maximum against the logarithm of the radius.
The result is a curve that bends upward, so between two radii it never rises above the straight line joining its values there.

### L2 human

The three-circle theorem is a quantitative form of the maximum modulus principle.
It is the circle version of the [[Hadamard three-lines theorem|WHS58R43N0]]; the map $z \mapsto e^{z}$ carries a vertical strip onto an annulus.
It is used in the theory of entire functions to study the growth of $M(r)$, and it extends to subharmonic functions.