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

摘要

Let (Gamma) and (psi=rac{Gamma'}{Gamma}) be respectively the classical Euler gamma function and the psi function and let (gamma=-psi(1)=0.57721566dotsc) 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].