This is a follow-up to the MathStackexchange question 3305 and MathOverflow question 23229.
The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies:
$(1)\ f(x+1) = xf(x), \ (2)\ f(1) = 1, \ (3)\ \ln \circ f \ $ is convex.
Assume a function $f\colon \mathbb{R^{+}} \to \mathbb{R}$ that satisfies:
$(1)\ f(x+1) = xf(x), \ (2)\ f(1) = 1, \ (3)\ $ f is superadditive.
Is there a (natural) condition which makes this function unique?
Edit 1: The answer is 'no' as Moron explains. And if we assume $f$ superadditive only for $x,y \ge a$ for some real $a$?
Edit 2: I accept Moron's answer because it the correct answer to my question. Intended was the question in the sense of my first edit (and my first question). I am also curious to see an answer if condition (3) reads: (3'') $\ \ln \circ f \ $ is superadditive. Thanks to whuber for suggesting this.