Bohr-Mollerup theorem
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In mathematical analysis, the Bohr-Mollerup theorem is named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it. The theorem characterizes the gamma function, defined for x > 0 by
- <math>\Gamma(x)=\int_0^\infty t^{x-1} e^{-t}\,dt<math>
as the only function f on the interval x > 0 that simultaneously has the three properties
- <math>f(1)=1,<math> and
- <math>f(x+1)=xf(x)\ \mbox{for}\ x>0,<math> and
- <math>\log f<math> is a convex function.
That log f is convex is often expressed by saying that f is log-convex, i.e., a log-convex function is one whose logarithm is convex.
External link
Proof, at PlanetMath (http://planetmath.org/?op=getobj&from=objects&id=3808)Template:Math-stub