Hadamard three-lines theorem
The Hadamard three-lines theorem is a result, in the branch of mathematics known as complex analysis, 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 of theorem
Let f(z) be a bounded function of z = x + iy defined on the strip
holomorphic in the interior of the strip and continuous on the whole strip. If
then log M(x) is a convex function on [a, b].
In other words, if with , then
Thus |F(z)| ≤ 1 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 z, it can be assumed that a = 0 and b = 1. The function
tends to 0 as |z| tends to infinity and satisfies |Fn| ≤ 1 on the boundary of the strip. The maximum modulus principle can therefore be applied to Fn in the strip. So |Fn(z)| ≤ 1. Since Fn(z) tends to F(z) as n tends to infinity. it follows that |F(z)| ≤ 1.
shows that, if
then is a convex function of s.
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
- Hadamard, Jacques (1896), "Sur les fonctions entières" (PDF), Bull.Soc. Math. France, 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, 97, American Mathematical Society, pp. 386–387, ISBN 0-8218-4479-2