In this paper, we present some completely monotonic functions and asymptotic expansions related to the gamma function. Based on the obtained expansions, we provide new bounds for Γ(x + 1)/Γ(x + 1/2) and Γ(x + 1/2).
Chen, C-P., & Paris, R. B. (2015). Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function. Applied Mathematics and Computation, 250, 514-529. https://doi.org/10.1016/j.amc.2014.11.010