# Bohr-Mollerup theorem

Let $f:\mathbb{R}^{+}\rightarrow\mathbb{R^{+}}$ be a function with the following properties:

1. 1.

$\log f(x)$ is a convex function (i.e. $f$ is logarithmically convex);

2. 2.

$f(x+1)=xf(x)$ for all $x>0$;

3. 3.

$f(1)=1.$

Then $f(x)=\Gamma(x)$ for all $x>0$.
That is, the only function satisfying those properties is the gamma function (restricted to the positive reals.)

Title Bohr-Mollerup theorem BohrMollerupTheorem 2013-03-22 13:15:10 2013-03-22 13:15:10 Koro (127) Koro (127) 5 Koro (127) Theorem msc 33B15 characterization of the gamma function GammaFunction LogarithmicallyConvexFunction