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We prove: (i) A logarithmically completely monotonic function is completely monotonic. (ii) For $ x>0 $ and $ n=0, 1, 2, ldots $, then$$ (-1)^{n}left(ln frac{x Gamma(x)}{sqrt{x+1/4},Gamma(x+1/2)}ight)^{(n)}>0. $$(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.</p

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