Top Qs
Timeline
Chat
Perspective
Hadamard three-lines theorem
From Wikipedia, the free encyclopedia
Remove ads
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
Summarize
Perspective
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
Remove ads
Applications
Summarize
Perspective
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
Remove ads
See also
References
- 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
Remove ads
Wikiwand - on
Seamless Wikipedia browsing. On steroids.
Remove ads