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. Let "f"("z") be a bounded function of "z = x + iy" defined on the strip  holomorphic in the interior of the strip and continuous on the whole strip. If  then log "M"("x") is a convex function on ["a", "b"]. In other words, if formula_3 with formula_4, then  Define formula_6 by  Thus |"F"("z")| ≤ 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  tends to 0 as |"z"| tends to infinity and satisfies |"F""n"| ≤ 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")| ≤ 1. Since "F""n"("z") tends to "F"("z") as "n" tends to infinity. it follows that |"F"("z")| ≤ 1. The three-line theorem can be used to prove the Hadamard three-circle theorem for a bounded continuous function formula_9 on an annulus formula_10, holomorphic in the interior. Indeed applying the theorem to  shows that, if  then formula_13 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  where formula_15, by considering the function