---
tag: WHS58R43N0
kind: statement
role: theorem
name: Hadamard three-lines theorem
msc: [30C80, 30A10]
uses: [0DG0Z72CQD, 048F2P90WR, 0BTMQ04556]
about: [0NAHXR1FRH]
sources:
  - url: https://en.wikipedia.org/wiki/Hadamard_three-lines_theorem
same_as:
  wikidata: Q3527250
  wikipedia: Hadamard three-lines theorem
---

## Statement

Let $S = \set{z \in \C : 0 \le \operatorname{Re} z \le 1}$ be the closed vertical strip.
Let $f \colon S \to \C$ be continuous and bounded on $S$ and [[holomorphic|0DG0Z72CQD]] on the interior of $S$.
For $0 \le x \le 1$ let $M(x) = \sup_{y \in \R} \abs{f(x + iy)}$.
Then for each $x$ with $0 \le x \le 1$,
$$M(x) \le M(0)^{1 - x} M(1)^{x}.$$

## Remarks

Proof idea: assume $M(0), M(1) > 0$ and put $g(z) = f(z) M(0)^{z - 1} M(1)^{-z}$, so that $\abs{g} \le 1$ on both boundary lines.
For $\varepsilon > 0$ the function $g(z) e^{\varepsilon z^2}$ tends to $0$ as $\abs{\operatorname{Im} z} \to \infty$ in $S$, since $g$ is bounded.
The [[maximum modulus principle|048F2P90WR]] on long rectangles gives $\abs{g(z)} e^{\varepsilon (x^2 - y^2)} \le e^{\varepsilon}$, and $\varepsilon \to 0$ gives $\abs{g} \le 1$.
The boundedness hypothesis is needed.
The function $f(z) = \exp(i e^{-i\pi z})$ satisfies $\abs{f(x + iy)} = \exp(\sin(\pi x) e^{\pi y})$.
It has modulus $1$ on both lines but is unbounded on the line $\operatorname{Re} z = \frac{1}{2}$.
This theorem is the key step in the Riesz–Thorin interpolation theorem.

## Explanations

### L1 human

Look at a holomorphic function on a vertical strip, and record its largest size on each vertical line.
The theorem says that the logarithm of this size is a convex function of the position across the strip.
So the size in the middle is controlled by the sizes on the two edges, in a geometric-mean way.

### L2 human

The theorem is the strip form of the [[Hadamard three-circle theorem|V00Q05ZW1G]] and a special case of the [[Phragmén–Lindelöf principle|FYR6V2NH6C]].
Its main use is in interpolation: the Riesz–Thorin theorem bounds an operator on $L^{p}$ for intermediate $p$ by applying it to an analytic family of functions.
The same idea appears in complex interpolation of Banach spaces and in convexity bounds for the Riemann zeta function in the critical strip.