Complete monotonicity of the entropy in the central limit theorem for gamma and inverse Gaussian distributions

Let Hg(α) be the differential entropy of the gamma distribution Gam(α,α). It is shown that (1/2)log(2πe)−Hg(α) is a completely monotone function of α. This refines the monotonicity of the entropy in the central limit theorem for gamma random variables. A similar result holds for the inverse Gaussian family. How generally this complete monotonicity holds is left as an open problem.