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.
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.
log
is a convex function on
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
tends to infinity and satisfies
on the boundary of the strip.
The maximum modulus principle can therefore be applied to
in the strip.
tends to infinity, it follows that
∎ The three-line theorem can be used to prove the Hadamard three-circle theorem for a bounded continuous function
holomorphic in the interior.
Indeed applying the theorem to shows that, if then
log
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