Sharp inequalities and complete monotonicity for the Wallis ratio (Q617786)

If \(x>0\) and \(a\geq 0\) let \(f_a(x)=x! -(x-{1\over2})! -\log\sqrt{x+a}\). The author gives an elementary proof that if \(0\leq a\leq {1\over4}\) then \(f_a(x)\) is completely monotonic, while if \(a\geq {1\over2}\) then \(-f_a(x)\) is completely monotonic. Using this he then deduces that: \[ \sqrt{x+{1\over4}}<{x!\over(x-{1\over2})!}<{4\over\sqrt{5 \pi}}\sqrt{x+{1\over4}}, \quad\text{and}\quad 2\sqrt{{2\over3 \pi}}\sqrt{x+{1\over2}}<{x!\over(x-{1\over2})!}<\sqrt{x+{1\over2}} , \] where the constants being best possible. Other inequalities involving the digamma function are also obtained.