Supplements to a class of logarithmically completely monotonic functions associated with the gamma function

In this article, the authors present a necessary condition, a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic. As a consequence of these results, supplements to the recent investigation by Chen and Qi [C.-P. Chen, F. Qi, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (2006) 405-411] are provided and a new Kečkić-Vasić type inequality is concluded.