Hadamard three-lines theorem

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