The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies:
1) f(x+1) = x*f(x)
2) f(1) = 1
3) ln(f(x)) is convex
The Gamma function is meant to interpolate the factorial function, so I can see the importance of the first two properties. But why is log convexity important? How does it affect the Gamma function's applicability in other areas of mathematics?
First, let me mention that log convexity of a function is implied by an analytic property, which appears to be more natural than log convexity itself. Namely, if $\mu$ is a Borel measure on $[0,\infty)$ such that the $r$th moment $$f(r)=\int_{0}^{\infty}z^r d\mu(z)$$ is finite for all $r$ in the interval $I\subset \mathbb R$, then $\log f$ is convex on $I$.
Log convexity can be effectively used in derivation of various inequalities involving the gamma function (particularly, two-sided estimates of products of gamma functions). It is linked with the notion of Schur convexity which is itself used in many applications.
An appetizer. Let $m=\max x_i$, $s=\sum x_i$, $x_i > 0$, $i = 1,\dots,n$, then $$[\Gamma(s/n)]^n\leq\prod\limits_{1}^{n}\Gamma (x_i)\leq \left[\Gamma\left(\frac{s-m}{n-1}\right)\right]^{n-1}\Gamma(m).\qquad\qquad\qquad (1)$$
(1) is trivial, of course, when all $x_i$ and $s/n$ are integers, but in general the bounds do not hold without assuming log convexity.
Edit added: a sketch of the proof. Let $f$ be a continuous positive function defined on an interval $I\subset \mathbb R$. One may show that the function $\phi(x)=\prod\limits_{i=1}^{n}f(x_i)$, $x\in I^n$ is Schur-convex on $I^n$ if and only if $\log f$ is convex on $I$. Thus the function $$\phi(x)=\prod\limits_{i=1}^n \Gamma(x_i),\quad x_i>0,\qquad \quad\qquad\qquad\qquad\qquad\qquad\quad (2)$$ is Schur-convex on $I^n=(0,\infty)^n$. Since $x_i\le m$, $i=1,\dots,n$, and $\sum x_i=s$, it is easy to check that $$x \prec \left(\frac{s-m}{n-1},\dots,\frac{s-m}{n-1},m\right).$$ The latter majorization and the fact that $\phi(x)$ defined by (2) is Schur-convex imply the upper bound (1). The lower bound follows from the standard majorization $x\succ (s/n,\dots,s/n)$.
Have a look at the recent short article by Marshall and Olkin concerning this and related inequalities for the gamma function.
There is a lovely and detailed discussion of the uniqueness of the gamma function in Volume 2 of Markushevich's Theory of Functions of a Complex Variable. The essential points are as follows.
Any meromorphic solution $f$ to the functional equation $zf(z)=f(z+1)$, with $f(1)=1$, must have simple poles at all integers $m\le 0$ with residues $(-1)^m/m!$ Assuming that these are ALL the poles of $f$, Markushevich exhibits all meromorphic interpolations of the factorial function. It turns out that there is a whole zoo of them, and in some (not very satisfying) sense the usual gamma function is the simplest choice.
So ... The Bohr-Mollerup Theorem sheds some light on this situation. But maybe there are other "interesting" extensions of the factorial that are not log-convex.
