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 (that have convexity or concavity properties of any order).

Statement
Bohr–Mollerup Theorem.      is the only function that satisfies  with  convex and also with .

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 (which preserves the inequalities since the exponential function is monotonically increasing) 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 .  And the final critical part of the proof is to remember that the limit of a sequence is unique.  This means that for any choice of  only one possible number  can exist.  Therefore, there is no other function 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.

See also 
 Wielandt theorem

References

Gamma and related functions
Theorems in complex analysis