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 , 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.