Аннотация:
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.
Ключевые слова:
Stirling's formula, gamma and polygamma functions, Laplace transforms, complete monotonicity and strongly complete monotonicity.
Список цитирования
