Complete monotonicity of the remainder of the asymptotic series for the ratio of two gamma functions

Abstract

For u>0 with u≠1,2 and m∈N and letfm(x;u)=1u−1ln⁡Γ(x+u)Γ(x+1)−ln⁡(x+u2)−∑k=2mck(u)xk. We prove the complete monotonicity of fm(x;u) for m=2,3 and f4n−1(x;1/2), −f4n−2(x;1/2) for n∈N in x on (0,∞). From these several new sharp bounds for the ratio of gamma functions and divided difference of polygamma functions are established. Lastly, a conjecture is posed.