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. Define by where 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 it can be assumed that and The function tends to as tends to infinity and satisfies on the boundary of the strip. The maximum modulus principle can therefore be applied to in the strip. So Because tends to as 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 on an annulus holomorphic in the interior. Indeed applying the theorem to shows that, if then 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.