Optimization Seminar

Title: Convexity in the uniqueness of the Gamma function
Speaker: Michael Orlitzky

Abstract:
The function Gamma(x) defined as the integral (with respect to t) of t^{x-1}e^{-t} from zero to infinity is the unique positive function f on (0, infinity) possessing the following three properties:

1. f(x+1) = xf(x)
2. f(1) = 1
3. f is log-convex

The proof of this fact involves two applications of convexity. First, Holder's inequality is used to prove that Gamma is log-convex, appealing to the fact that (1/p)x + (1/q)y is a convex combination. Then, the fact that Gamma is determined recursively means that we only need to show uniqueness on (0,1). This lets us write a convex combination of n and n+1 as n+x, and we use an inequality for convex functions to derive an otherwise-difficult result.