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 Γ(1+w)∼2πw(w/e)ww2+760w2−120w/2exp∑r=1∞μrwr,w→∞, with the recurrence relation of coefficients μr. Furthermore, we use Padé approximants and our new asymptotic expansion to deduce the new bounds of Γ(w) better than some of its recent ones.
The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function Γμ(v) and study the completely monotonic (CM) property of a function involving Γμ(v) and the generalized digamma function ψμ(v). As a consequence, we establish some bounds for Γμ(v), ψμ(v) and polygamma functions ψμ(r)(v), r≥1.
In this paper, we study the completely monotonic property of two functions involving the function (x) = [ψ (x)] 2 + ψ (x) and deduce the double inequalitywhich improve some recent results, where ψ(x) is the logarithmic derivative of the Gamma function. Also, we deduce the completely monotonic degree of a function involving ψ (x).
Two new approximation formulas for Bateman’s G-function are presented with strictly monotonic error functions and we deduced their sharp bounds. We also studied the completely monotonic (CM) degrees of two functions involving G(r), deducing two of its inequalities and improving some of the recently published results.
We prove that the function σ(s) defined by β(s)=6s2+12s+53s2(2s+3)−ψ′(s)2−σ(s)2s5,s>0, is strictly increasing with the sharp bounds 0<σ(s)<49120, where β(s) is Nielsen’s beta function and ψ′(s) is the trigamma function. Furthermore, we prove that the two functions s↦(−1)1+μβ(s)−6s2+12s+53s2(2s+3)+ψ′(s)2+49μ240s5, μ=0,1 are completely monotonic for s>0. As an application, double inequality for β(s) involving ψ′(s) is obtained, which improve some recent results.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.