Hadamard three-lines theorem

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.

Statement
$$

Define $$ F(z)$$ by


 * $$ F(z)=f(z) M(a)^M(b)^$$

where $$|F(z)| \leq 1$$ 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


 * $$ F_n(z) = F(z) e^{z^2/n}e^{-1/n} $$

tends to $$0$$ as $$|z|$$ tends to infinity and satisfies $$|F_n| \leq 1$$ on the boundary of the strip. The maximum modulus principle can therefore be applied to $$F_n$$ in the strip. So $$|F_n(z)| \leq 1.$$ Because $$F_n(z)$$ tends to $$F(z)$$ as $$n$$ tends to infinity, it follows that $$|F(z)| \leq 1.$$ ∎

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


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