Hadamard Three-line 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

:$F_n(z) = F(z) e^e^$

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 $\,$ holomorphic in the interior. Indeed applying the theorem to

:$f(z) = g(e^),$

shows that, if

:$m(s) = \sup_ , g(z), ,$

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

:$\int , gh, \leq \left(\int , g, ^p\right)^ \cdot \left(\int , h, ^q\right)^,$

where $+ = 1,$ by considering the function

:$f(z) = \int , g, ^ , h, ^.$

See also

* Riesz–Thorin theorem
* Phragmén–Lindelöf principle

References

* (the original announcement of the theorem)
*
* {{citation, title=Complex made simple, volume= 97, series= Graduate Studies in Mathematics, first= David C., last= Ullrich, publisher=American Mathematical Society, year= 2008, isbn=978-0-8218-4479-3, pages=386–387

Convex analysis
Theorems in complex analysis