Inequalities for the beta function

Abstract

Let g(x):= (e/x)xΓ(x+1) and F(x,y):= g(x)g(y)/g(x+y). Let Dx,y(k) be the k th differential in Taylor's expansion of logF(x,y) . We prove that when (x,y) ∈ R+2 one has (-1)k-1Dx,y(k) (X,Y) > 0 for every X,Y ∈ R+, and that when k is even the conclusion holds for every X,Y ∈ R with (X,Y) = (0,0). This implies that Taylor's polynomials for logF provide upper and lower bounds for logF according to the parity of their degree. The formula connecting the Beta function to the Gamma function shows that the bounds for F are actually bounds for Beta.