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 x > 0 by

${\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,\mathrm {d} t}$

as the only positive function  f , with domain on the interval x > 0, that simultaneously has the following three properties:

•  f (1) = 1, and
•  f (x + 1) = x f (x) for x > 0 and
•  f  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).