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.