Hadamard three-lines theorem

The Hadamard three-lines theorem is a result, in the branch of mathematics known as complex analysis, 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 ${\displaystyle x=ta+(1-t)b}$ with ${\displaystyle 0\leq t\leq 1}$, then

Define ${\displaystyle F(z)}$ by

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