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

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

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and

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and

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is a convex function. (That is Failed to parse (Missing texvc executable; please see math/README to configure.): f \,
is logarithmically convex.)

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.

An elegant 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.

References

• Eric W. Weisstein, Bohr-Mollerup Theorem at MathWorld.
• Proof of Bohr-Mollerup theorem on PlanetMath
• Proof of Bohr-Mollerup theorem on PlanetMath
• Artin, Emil (1964). The Gamma Function. Holt, Rinehart, Winston. 
• Rosen, Michael (2006). Exposition by Emil Artin: A Selection. American Mathematical Society. 

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