# Ngô Quốc Anh

## September 29, 2010

### The Three Lines Theorem by Hadamard

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 11:56

Theorem. Let $\phi(\xi)$ be a bounded analytic function in the strip $0 \leqslant {\rm Re } \, \xi \leqslant 1$. Denote

$\displaystyle N(a)=\sup_\eta|\phi (a+i\eta)|$.

Then

$\displaystyle N(a) \leqslant N^{1-a}(0)N^a(1)$

Proof. Set $c=\log \frac{N(0)}{N(1)}$. By the hypothesis, the function $\phi(\xi)e^{c\xi}$ is in absolute value $\leqslant N(0)$ for ${\rm Re } \, \xi=0$ and ${\rm Re } \, \xi=1$. So, by the maximum principle applied in the strip $0\leqslant {\rm Re } \, \xi\leqslant 1$,

$\displaystyle |\phi(a+i\eta)|e^{ca} \leqslant N(0)$;

from this and the definition of $c$, the inequality follows.

Source: Functional Analysis by Peter Lax.