On Stirling's formula remainder

Let the sequence rn be defined byn!=2πnn/enern.We establish new estimates for Stirling's formula remainder rn. This improves some known results. Let θ(x) be defined by the relation:Γ(x+1)=2π(x/e)xeθ(x)/(12x).We prove that θ″(x) is completely monotonic on (0,∞). This implies the result given by Mortici, who proved that -x-1θ‴(x) is completely monotonic on (0,∞).