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

✍ Scribed by Cristinel Mortici

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 193 KB
• Volume: 24
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2010.12.034

✦ Synopsis

proved that the function θ (x) defined by Γ (x + 1) = √ 2π (x/e) x e θ (x)/12x 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 . . . .