Bohr-Mollerup theorem

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

1. $\log f(x)$ 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.)