Bohr–Mollerup theorem

Remarks

The gamma function satisfies the three conditions; its log-convexity follows from Hölder's inequality applied to $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t}\, dt$. For uniqueness, conditions 1 and 2 give $f(n + 1) = n!$, so the values at integers are fixed. Fix $0 < x \le 1$ and an integer $n \ge 2$. Convexity of $\ln f$ compares its slope on $[n, n + x]$ with its slopes on $[n - 1, n]$ and $[n, n + 1]$. This squeezes $f(n + x)$ between two explicit bounds, and condition 2 turns them into bounds for $f(x)$. Letting $n \to \infty$ yields $f(x) = \lim_{n \to \infty} \frac{n!\, n^x}{x (x + 1) \cdots (x + n)}$, the same limit for every such $f$. Without condition 3 uniqueness fails: $f(x) = \Gamma(x)\,(1 + \frac12 \sin(2\pi x))$ satisfies conditions 1 and 2, takes positive values, and differs from $\Gamma$ at $x = \frac14$.

Going deeper

For a student

The theorem is a convenient way to prove identities for $\Gamma$: show that the other side satisfies the three conditions. For example, it gives the formula $B(x, y) = \Gamma(x)\Gamma(y)/\Gamma(x + y)$ for the beta function, by applying the theorem to $x \mapsto B(x, y)\,\Gamma(x + y)/\Gamma(y)$ for fixed $y$. It also gives the Gauss product formula for $\Gamma$ and the Legendre duplication formula. It links the theory of convex functions with special functions.