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

Abstract

Let H g ( α ) be the differential entropy of the gamma distribution Gam ( α , α ) . It is shown that ( 1 / 2 ) log ( 2 π e ) − H g ( 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.