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?

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See this. –  Neves Mar 13 '11 at 9:46
For completeness, this is the Bohr–Mollerup theorem. –  lhf Apr 12 '11 at 10:48
Artin's book The Gamma Function contains some other results on uniqueness that depend on continuity or continuous differentiability only but require Legendre functional equation. –  lhf Apr 12 '11 at 10:59