Bohr–Mollerup theorem

In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for by
as the only positive function, with domain on the interval, that simultaneously has the following three properties:

, and
for and
is logarithmically convex.

A treatment of this theorem is in Artin's book The Gamma Function, which has been reprinted by the AMS in a collection of Artin's writings.
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
The theorem admits a far-reaching generalization to a wide variety of functions.

Proof

Let be a function with the assumed properties established above: and is convex, and. From we can establish
The purpose of the stipulation that forces the property to duplicate the factorials of the integers so we can conclude now that if and if exists at all. Because of our relation for, if we can fully understand for then we understand for all values of.
For,, the slope of the line segment connecting the points and is monotonically increasing in each argument with since we have stipulated that is convex. Thus, we know that
After simplifying using the various properties of the logarithm, and then exponentiating we obtain
From previous work this expands to
and so
The last line is a strong statement. In particular, it is true for all values of. That is is not greater than the right hand side for any choice of and likewise, is not less than the left hand side for any other choice of. Each single inequality stands alone and may be interpreted as an independent statement. Because of this fact, we are free to choose different values of for the RHS and the LHS. In particular, if we keep for the RHS and choose for the LHS we get:
It is evident from this last line that a function is being sandwiched between two expressions, a common analysis technique to prove various things such as the existence of a limit, or convergence. Let :
so the left side of the last inequality is driven to equal the right side in the limit and
is sandwiched in between. This can only mean that
In the context of this proof this means that
has the three specified properties belonging to. Also, the proof provides a specific expression for, that must hold if the characterizing properties are true. Therefore, there is no other function defined on with all the properties assigned to.
The remaining loose end is the question of proving that makes sense for all where
exists. The problem is that our first double inequality
was constructed with the constraint. If, say, then the fact that is monotonically increasing would make, contradicting the inequality upon which the entire proof is constructed. However,
which demonstrates how to bootstrap to all values of where the limit is defined.