A Note on Complete Monotonicity of the Remainder in Stirling's Formula

Mortici [C. Mortici, “On the monotonicity and convexity of the remainder of the Stirling formula,” Appl. Math. Lett. 24 (6), 869–871 (2011)] showed that the function $-x^{-1}\theta^{\prime\prime\prime}(x)$, where $\theta(x)$ is given by
$$ \Gamma(x+1)=\sqrt{2\pi}\biggl(\frac{x}{e}\biggr)^{x} e^{\theta(x)/{12x}}=\sqrt{2\pi x}\biggl(\frac{x}{e}\biggr)^{x}e^{\sigma(x)/{12x}} $$
is strictly completely monotonic on $(0,\infty)$. The aim of this paper is to prove that $\sigma^{\prime\prime\prime}(x)$ is strictly completely monotonic on $(0,\infty)$ by using the theory of Laplace transforms.