In the article, the sufficient and necessary conditions such that a
class of functions which involve the psi function ψ and the ratio (Γ(x+t))/(Γ(x+s)) are
logarithmically completely monotonic are established, the best bounds for the
ratio (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}.