A class of logarithmically completely monotonic functions and the best bounds in the second Kershawʹs double inequality

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 } .