The Bohr-Mollerup Theorem states that the gamma function is the unique function $f: (0, \infty )\rightarrow \mathbb{R}$ satisfying $f(1)=1,$ $f(x+1) = x f(x),$ and the condition that $\log f$ is convex. Logarithmically convex seems like a rather stringent imposition; what if $f$ is just convex? I'm not really sure how to devise a counterexample here, but I'm sure one must exist (otherwise the theorem would read prettier.) Anybody know anything about this?
Convex Alternatives to the Gamma Function?
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real-analysis
convex-analysis
gamma-function
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0After posting this question I came across his [page](http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunction.html) on the gamma function. Very interesting stuff. – 2012-06-17
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Construct a function $f$ with $f(x)=1,1 \leq x \leq 2$ with $f(x+1)=xf(x)$,then it is not difficult to find out that $f$ possess an increasing left derivative(not strictly increasing).So $f$ is a convex function. But $f$ is not a logarithmically convex function.
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1I see, I thought it was $(0,3)$ - so the origin is $(1,0)$, in any case it is misleading. – 2012-05-25