proof of Bohr-Mollerup theorem

To show that the gamma functionDlmfDlmfMathworldPlanetmath is logarithmically convex, we can examine the product representation:


Since this product converges absolutely for x>0, we can take the logarithmMathworldPlanetmath term-by-term to obtain


It is justified to differentiate this series twice because the series of derivatives is absolutely and uniformly convergent.


Since every term in this series is positive, Γ is logarithmically convex. Furthermore, note that since each term is monotonically decreasing, logΓ is a decreasing function of x. If x>m for some integer m, then we can bound the series term-by-term to obtain


Therefore, as x, d2Γ/dx20.

Next, let f satisfy the hypotheses of the Bohr-Mollerup theoremMathworldPlanetmath. Consider the functionMathworldPlanetmath g defined as eg(x)=f(x)/Γ(x). By hypothesis 3, g(1)=0. By hypothesis 2, eg(x+1)=eg(x), so g(x+1)=g(x). In other , g is periodic.

Suppose that g is not constant. Then there must exist points x0 and x1 on the real axis such that g(x0)g(x1). Suppose that g(x1)>g(x0) for definiteness. Since g is periodic with period 1, we may assume without loss of generality that x0<x1<x0+1. Let D2 denote the second divided differenceDlmfMathworldPlanetmath of g:


By our assumptions, D2<0. By linearity,


By periodicity, we have


for every integer n>0. However,


As n, the right hand side approaches zero. Hence, by choosing n sufficiently large, we can make the left-hand side smaller than |D2|/2. For such an n,


However, this contradicts hypothesis 1. Therefore, g must be constant. Since g(0)=0, g(x)=0 for all x, which implies that e0=f(x)/Γ(x). In other words, f(x)=Γ(x) as desired.

Title proof of Bohr-Mollerup theorem
Canonical name ProofOfBohrMollerupTheorem
Date of creation 2013-03-22 14:53:39
Last modified on 2013-03-22 14:53:39
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 18
Author Andrea Ambrosio (7332)
Entry type Proof
Classification msc 33B15