Convex Alternatives to the Gamma Function?

Question

The Bohr-Mollerup Theorem states that the gamma function is the unique function $f: (0, \infty )\rightarrow \mathbb{R}$ satisfying $f(1)=1,$ $f(x+1) = x f(x),$ and the condition that $\log f$ is convex. Logarithmically convex seems like a rather stringent imposition; what if $f$ is just convex? I'm not really sure how to devise a counterexample here, but I'm sure one must exist (otherwise the theorem would read prettier.) Anybody know anything about this?

Answer 1

Construct a function $f$ with $f(x)=1,1 \leq x \leq 2$ with $f(x+1)=xf(x)$,then it is not difficult to find out that $f$ possess an increasing left derivative(not strictly increasing).So $f$ is a convex function. But $f$ is not a logarithmically convex function.