The uniqueness of the Gamma Function

It is a theorem that any function $f$ defined for positive real numbers satisfying

1. $f(1)=1$
2. $f(x+1)=x\cdot f(x)$
3. $f$ is log convex

is identically equal to the gamma function. (Condition 2 means that this function interpolates a shifted factorial function.)

Now, a beginner (such as myself) might ask: What if we weaken condition 2 by instead requiring $f$ to be merely convex, not log convex?

I would imagine that such functions would look not too different, since intuitively, I can't wildly deviate the graph of the gamma function if I want to maintain condition 2 and stay convex.

Just a follow-up musing---What if instead of condition 3, we require convexity and infinite differentiability? Do we still uniquely determine the gamma function?

Solution 1:

(This should be either a comment or CW.)

Peter Luschny studied a number of gamma-like functions that do not have the log-convex imposition; you might want to look into them for inspiration.