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

Template:Collapse top Define $F(z)$ by

$F(z)=f(z)M(a{)}^{\frac{z-b}{b-a}}M(b{)}^{\frac{z-a}{a-b}}$

where $|F(z)|\le 1$ 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

$F_n(z)=F(z)e^{z^2/n}{e}^{-1/n}$

tends to $0$ as $|z|$ tends to infinity and satisfies $|F_n|\le 1$ on the boundary of the strip. The maximum modulus principle can therefore be applied to $F_n$ in the strip. So $|F_n(z)|\le 1.$ Because $F_n(z)$ tends to $F(z)$ as $n$ tends to infinity, it follows that $|F(z)|\le 1.$ ∎ Template:Collapse bottom

Applications

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

$f(z)=g(e^z),$

shows that, if

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

then $\log m(s)$ 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

$\int |gh|\le {(\int |g{|}^p)}^{\frac{1}{p}}\cdot {(\int |h{|}^q)}^{\frac{1}{q}},$

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

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

See also

• Riesz–Thorin theorem
• Phragmén–Lindelöf principle

References

• Template:Citation (the original announcement of the theorem)
• Template:Citation
• Template:Citation