Completely monotonic function associated with the Gamma functions and proof of Wallis' inequality

Abstract

We prove: (i) A logarithmically completely monotonic function is completely monotonic. (ii) For $ x>0 $ and $ n=0, 1, 2, \ldots $, then

$$ (-1)^{n}\left(\ln \frac{x \Gamma(x)}{\sqrt{x+1/4}\,\Gamma(x+1/2)}\right)^{(n)}>0. $$

(iii) For all natural numbers $ n $, then

$$ \frac1{\sqrt{\pi(n+4/ \pi-1)}}\leq \frac{(2n-1)!!}{(2n)!!}<\frac1{\sqrt{\pi(n+1/4)}}. $$

The constants $ \frac{4}{\pi}-1$ and $\frac14 $ are the best possible.