A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

<p><span>In the paper, the authors find necessary conditions and sufficient conditions for a function involving the gamma function and originating from investigation of properties of the Catalan numbers and function in combinatorics to be logarithmically completely monotonic.</span></p>

Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions

Let \(\Gamma\) and \(\psi=\frac{\Gamma'}{\Gamma}\) be respectively the classical Euler gamma function and the psi function and let \(\gamma=-\psi(1)=0.57721566\dotsc\) stand for the Euler-Mascheroni constant. In the paper, the authors simply confirm the logarithmically complete monotonicity of the power-exponential function \(q(t)=t^{t[\psi(t)-\ln t]-\gamma}\) on the unit interval \((0,1)\), concisely deny that \(q(t)\) is a Stieltjes function, surely point out fatal errors appeared in the paper [V. Krasniqi and A. Sh. Shabani, On a conjecture of a logarithmically completely monotonic function, Aust. J. Math. Anal. Appl. 11 (2014), no.1, Art.5, 5 pages; Available online at http://ajmaa.org/cgi-bin/paper.pl?string=v11n1/V11I1P5.tex], and partially solve a conjecture posed in the article [B.-N. Guo, Y.-J. Zhang, and F. Qi, Refinements and sharpenings of some double inequalities for bounding the gamma function, J. Inequal. Pure Appl. Math. 9 (2008), no.1, Art.17; Available online at http://www.emis.de/journals/JIPAM/article953.html].

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

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)!!}