Abstract
Inequalities play a fundamental role in both theoretical and applied mathematics and contain many patterns of symmetries. In many studies, inequalities have been used to provide estimates of some functions based on the properties of their symmetry. In this paper, we present the following new asymptotic expansion related to the ordinary gamma function Gamma(1 + w) similar to root 2 pi w(w/e)(w) (w(2) + 7/60/w(2)- 1/20)(w/2) exp (Sigma(infinity)(r=1) mu(r)/w(r)), w -> infinity, with the recurrence relation of coefficients mu(r). Furthermore, we use Fade approximants and our new asymptotic expansion to deduce the new bounds of Gamma(w) better than some of its recent ones.