On the monotonicity and convexity of the remainder of the Stirling formula

AbstractShi, Liu and Hu [X. Shi, F. Liu, M. Hu, A new asymptotic series for the Gamma function, J. Comput. Appl. Math. 195 (2006) 134–154] proved that the function θ(x) defined by Γ(x+1)=2π(x/e)xeθ(x)/12x is strictly increasing for x≥1. The aim of our work is to prove that −x−1θ‴(x) is strictly completely monotonic on (0,∞). As direct consequences, we show that θ is strictly convex on (0,∞), and then we prove that θ is strictly decreasing on (0,β), and strictly increasing on (β,∞), where β=0.34142…