A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality

In the article, the sufficient and necessary conditions such that a class of functions which involve the psi function ψψ and the ratio Γ(x+t)/Γ(x+s)Γ(x+t)/Γ(x+s) are logarithmically completely monotonic are established, the best bounds for the ratio Γ(x+t)/Γ(x+s)Γ(x+t)/Γ(x+s) are given, and some comparisons with known results are carried out, where s and t   are two real numbers and x>-min{s,t}x>-min{s,t}.