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.

Statement[]

Hadamard three-lines theorem — Let $f(z)$ be a bounded function of $z=x+iy$ defined on the strip

\{x+iy:a\leq x\leq b\},

holomorphic in the interior of the strip and continuous on the whole strip. If

M(x)=\sup _{y}|f(x+iy)|

then $\log M(x)$ is a convex function on $[a,b].$

In other words, if $x=ta+(1-t)b$ with $0\leq t\leq 1,$ then

M(x)\leq M(a)^{t}M(b)^{1-t}.

Proof

Define $F(z)$ by

F(z)=f(z)M(a)^{z-b \over b-a}M(b)^{z-a \over a-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)}.

References[]

Hadamard, Jacques (1896), "Sur les fonctions entières" (PDF), Bull. Soc. Math. Fr., 24: 186–187 (the original announcement of the theorem)
Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics, Volume 2: Fourier analysis, self-adjointness, Elsevier, pp. 33–34, ISBN 0-12-585002-6
Ullrich, David C. (2008), Complex made simple, Graduate Studies in Mathematics, vol. 97, American Mathematical Society, pp. 386–387, ISBN 0-8218-4479-2

Hadamard three-lines theorem

Contents

Statement[]

Applications[]

See also[]

References[]