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.

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