A completely monotone function related to the Gamma function

摘要

We show that the reciprocal of the functionf(z)=logΓ(z+1)zlogz,z∈C⧹]−∞,0]is a Stieltjes transform. As a corollary we obtain that the derivative of f is completely monotone, in the sense that (−1)n−1f(n)(x)⩾0 for all n⩾1 and all x>0. This answers a question raised by Dimitar Dimitrov at the Fifth International Symposium on Orthogonal Polynomials, Special Functions and Applications held in Patras in September 1999. To prove the result we examine the imaginary part of 1/f in the upper half-plane, in particular close to the negative real axis, where Stirling's formula is not valid.