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

Shi, 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 ) = 2 π ( x / e ) x e θ ( x ) / 12 x is strictly increasing for x ≥ 1 . The aim of our work is to prove that − x − 1 θ ‴ ( 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 … .