We prove this theorem in two stages: first, we establish that the gamma function satisfies the given conditions and then we prove that these conditions uniquely determine a function on $(0,\infty)$ .

By its definition, $\Gamma(x)$ is positive for positive $x$ . Let $x,y>0$ and $0\leq \lambda \leq 1$ .

\begin{eqnarray*} \log \Gamma(\lambda x+(1-\lambda)y) &=& \log \int_0^\infty e^{-t}t^{\lambda x+(1-\lambda)y-1}dt\\ &=&\log\int_0^\infty (e^{-t}t^{x-1})^\lambda (e^{-t}t^{y-1})^{1-\lambda}dt\\ &\leq&\log ((\int_0^\infty e^{-t}t^{x-1}dt)^\lambda(\int_0^\infty e^{-t}t^{y-1}dt)^{1-\lambda})\\ &=& \lambda\log\Gamma(x)+(1-\lambda)\log\Gamma(y) \end{eqnarray*} The inequality follows from Hölder's inequality, where $p=\frac{1}{\lambda}$ and $q=\frac{1}{1-\lambda}$ .

This proves that $\Gamma$ is log-convex. Condition 2 follows from the definition by applying integration by parts. Condition 3 is a trivial verification from the definition.

Now we show that the 3 conditions uniquely determine a function. By condition 2, it suffices to show that the conditions uniquely determine a function on $(0,1)$ .

Let $G$ be a function satisfying the 3 conditions, $0\leq x\leq 1$ and $n\in{\mathbb N}$ .

$n+x=(1-x)n+x(n+1)$ and by log-convexity of $G$ , $G(n+x)\leq G(n)^{1-x}G(n+1)^x=G(n)^{1-x}G(n)^xn^x=(n-1)!n^x$ .

Similarly $n+1=x(n+x)+(1-x)(n+1+x)$ gives $n!\leq G(n+x)(n+x)^{1-x}$ .

Combining these two we get

$$ n!(n+x)^{x-1}\leq G(n+x) \leq (n-1)!n^x $$

and by using condition 2 to express $G(n+x)$ in terms of $G(x)$ we find

$$ a_n:=\frac{n!(n+x)^{x-1}}{x(x+1)\dots(x+n-1)}\leq G(x) \leq \frac{(n-1)!n^x}{x(x+1)\dots(x+n-1)}=:b_n.$$

Now these inequalities hold for every positive integer $n$ and the terms on the left and right side have a common limit ($\lim_{n\rightarrow\infty}\frac{a_n}{b_n}=1$ ) so we find this determines $G$ .

As a corollary we find another expression for $\Gamma$ .

For $0\leq x\leq 1$ ,

$$ \Gamma(x)=\lim_{n\rightarrow\infty} \frac{n!n^x}{x(x+1)\dots(x+n)}.$$

In fact, this equation, called Gauß's product, goes for the whole complex plane minus the negative integers.