A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautschi–Kershaw’s inequality

In this article, the logarithmically complete monotonicity of the function [Γ(x+b)Γ(x+a)]1/(a−b)exp[ψ(x+c)] are discussed, where a,b,ca,b,c are real numbers and Γ is the classical Euler’s gamma function. From this, the best upper and lower bounds for Walls’ ratio Γ(x+1)Γ(x+s) are established, which refine the second Gautschi–Kershaw’s inequality.