The Three Lines Theorem by Hadamard | Ngô Quốc Anh

September 29, 2010

The Three Lines Theorem by Hadamard

Filed under: Uncategorized — Tags: three lines theorem — 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.

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