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Hadamard three-lines theorem

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In complex analysis, a branch of mathematics, the Hadamard three-line theorem is a result about the behaviour of holomorphic functions defined in regions bounded by parallel lines in the complex plane. The theorem is named after the French mathematician Jacques Hadamard.

Statement

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Hadamard three-line theorem — Let be a bounded function of defined on the strip

holomorphic in the interior of the strip and continuous on the whole strip. If

then is a convex function on

In other words, if with then

Proof

Define by

where on the edges of the strip. The result follows once it is shown that the inequality also holds in the interior of the strip. After an affine transformation in the coordinate it can be assumed that and The function

tends to as tends to infinity and satisfies on the boundary of the strip. The maximum modulus principle can therefore be applied to in the strip. So Because tends to as tends to infinity, it follows that

Applications

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The three-line theorem can be used to prove the Hadamard three-circle theorem for a bounded continuous function on an annulus holomorphic in the interior. Indeed applying the theorem to

shows that, if

then is a convex function of

The three-line theorem also holds for functions with values in a Banach space and plays an important role in complex interpolation theory. It can be used to prove Hölder's inequality for measurable functions

where by considering the function

See also

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References

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  • Hadamard, Jacques (1896), "Sur les fonctions entières" (PDF), Bull. Soc. Math. Fr., 24: 186–187 (the original announcement of the theorem)
  • Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics, Volume 2: Fourier analysis, self-adjointness, Elsevier, pp. 33–34, ISBN 0-12-585002-6
  • Ullrich, David C. (2008), Complex made simple, Graduate Studies in Mathematics, vol. 97, American Mathematical Society, pp. 386–387, ISBN 978-0-8218-4479-3