Logarithmically completely monotonic functions relating to the gamma function

In this paper, the logarithmically complete monotonicity of the function exΓ(x+β)/xx+β−α in (0,∞) for α∈R and β⩾0 is considered and the corresponding result by G.D. Anderson, R.W. Barnard, K.C. Richards, M.K. Vamanamurthy and M. Vuorinen is generalized. As applications of these results, some inequalities between identric mean and ratio of two gamma functions by J.D. Kečkić and P.M. Vasić are extended.