This post is a bit of a digression from recent area of focus–although (log-)convexity does come into play–but I wanted to give another proof of the famous result

Recall that we can define the gamma function on the positive real axis by the integral expression

This integral is indeed convergent since for all . The gamma function satisfies has a few nice properties, which turn out to uniquely characterize it.

- ;
- ;
- is log-convex.

*Proof.* We prove (1) by integration by parts. For , we have

(2) is immediate from evaluating the integral expression. For (3), observe that for , we have by the concavity of that

where the inequality follows from applying Hölder’s inequality.

We now prove the Bohr-Mollerup theorem, which uniquely characterizes the gamma function in terms of the three properties listed above. Note that the Bohr here is not the 20th century physicist Niels Bohr but rather his mathematician brother, Harald Bohr.

**Theorem. **(Bohr-Mollerup theorem)

Suppose satisfies

- for all $x>0$;
- ;
- is log-convex.

Then .

*Proof. *I claim that for all . Suppose equality holds for some . Then

Let and . Then by log-convexity,

We can get an estimate for by another application of the log-convexity of :

Since , we obtain the inequalities

Letting , we obtain from the squeeze theorem that

We now show that this identity holds for all and therefore is uniquely determined. Let such that . Then

Since the gamma function satisfies all three properties in the statement of the theorem, we obtain the identity

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