Completely monotonic function associated with the Gamma functions and proof of Wallis' inequality
2005 ◽
Vol 36
(4)
◽
pp. 303-307
◽
Keyword(s):
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)!!}
2015 ◽
Vol 5
(4)
◽
pp. 626-634
◽
2015 ◽
Vol 3
(4)
◽
pp. 140
◽
2010 ◽
Vol 14
(4)
◽
pp. 1623-1628
◽
2018 ◽
Vol 97
(3)
◽
pp. 453-458
2012 ◽
Vol 164
(7)
◽
pp. 971-980
◽
2012 ◽
Vol 12
(1)
◽
pp. 329-341
◽
2012 ◽
Vol 218
(19)
◽
pp. 9890-9897
◽
Keyword(s):