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

For u > 0 with u &NOTEQUexpressionL; 1, 2 and m is an element of N and let f(m) (x; u) = 1/u -1 In gamma(x + u)/gamma(x + 1) - ln (x + u/2) - sigma(m )(k=2)c(k) (u)/x(k) . We prove the complete monotonicity of f(m) (x; u) for m = 2, 3 and f(4n-1) (x; 1/2), -f(4n-2) (x; 1/2) for n is an element of N in x on (0, infinity). 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.(C) 2022 Elsevier Inc. All rights reserved.