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

1.
$\mathrm{log}f(x)$ is a convex function (i.e. $f$ is logarithmically convex);

2.
$f(x+1)=xf(x)$ for all $x>0$;

3.
$f(1)=1.$
Then $f(x)=\mathrm{\Gamma}(x)$ for all $x>0$.
That is, the only function satisfying those properties is the gamma function^{} (restricted to the positive reals.)
Title  BohrMollerup theorem^{} 

Canonical name  BohrMollerupTheorem 
Date of creation  20130322 13:15:10 
Last modified on  20130322 13:15:10 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  5 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 33B15 
Synonym  characterization of the gamma function 
Related topic  GammaFunction 
Related topic  LogarithmicallyConvexFunction 