# Bohr-Mollerup theorem

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

[itex]\Gamma(x)=\int_0^\infty t^{x-1} e^{-t}\,dt[itex]

as the only function f on the interval x > 0 that simultaneously has the three properties

• [itex]f(1)=1,[itex] and
• [itex]f(x+1)=xf(x)\ \mbox{for}\ x>0,[itex] and
• [itex]\log f[itex] 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.

Proof, at PlanetMath (http://planetmath.org/?op=getobj&from=objects&id=3808)Template:Math-stub

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