Hadamard three-lines theorem
# Hadamard three-lines theorem

In complex analysis, a branch of mathematics, the Hadamard three-lines 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-lines theorem — Let $f(z)$ be a bounded function of $z=x+iy$ defined on the strip

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

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

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

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

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

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

## Applications

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

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

shows that, if

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

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

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

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

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