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

Hadamard three-line theorem — Let ${\displaystyle f(z)}$ be a bounded function of ${\displaystyle z=x+iy}$ defined on the strip

${\displaystyle \{x+iy:a\leq x\leq b\},}$

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

${\displaystyle M(x)=\sup _{y}|f(x+iy)|}$

then ${\displaystyle \log M(x)}$ is a convex function on ${\displaystyle [a,b].}$

In other words, if ${\displaystyle x=ta+(1-t)b}$ with ${\displaystyle 0\leq t\leq 1,}$ then

${\displaystyle M(x)\leq M(a)^{t}M(b)^{1-t}.}$

Applications

The three-line theorem can be used to prove the Hadamard three-circle theorem for a bounded continuous function ${\displaystyle g(z)}$ on anannulus ${\displaystyle \{z:r\leq |z|\leq R\},}$ holomorphic in the interior. Indeed applying the theorem to

${\displaystyle f(z)=g(e^{z}),}$

shows that, if

${\displaystyle m(s)=\sup _{|z|=e^{s}}|g(z)|,}$

then ${\displaystyle \log \,m(s)}$ is a convex function of ${\displaystyle s.}$

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

${\displaystyle \int |gh|\leq \left(\int |g|^{p}\right)^{1 \over p}\cdot \left(\int |h|^{q}\right)^{1 \over q},}$

where ${\displaystyle {1 \over p}+{1 \over q}=1,}$ by considering the function

${\displaystyle f(z)=\int |g|^{pz}|h|^{q(1-z)}.}$

See also

• Riesz–Thorin theorem
• Phragmén–Lindelöf principle

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