It is common knowledge that the classical Euler gamma function $\Gamma(z)$ can defined by \begin{equation*} \Gamma(z)=\int^\infty_0t^{z-1} e^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and the psi (digamma) function is defined by $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$.
A non-negative function $f$ is said to be completely monotonic on an interval $I$ if $f$ has derivatives of all orders on $I$ and \begin{equation*} 0\le(-1)^{n-1}f^{(n-1)}(x)<\infty \end{equation*} for all $x\in I$ and $n\in\mathbb{N}=\{1,2,3,\dotsc\}$.
A positive function $f$ is said to be logarithmically completely monotonic on an interval $I$ if it is infinitely differentiable (smooth) and satisfies \begin{equation*} (-1)^k[\ln f(t)]^{(k)}\ge0 \end{equation*} on $I$ for $k\in\mathbb{N}=\{1,2,\dotsc\}$.
A logarithmically completely function on an interval $I$ must be also completely monotonic on $I$, but not conversely.
The Bernstein--Widder theorem reads that a function $f$ is completely monotonic on $(0,\infty)$ if and only if it can be represented as a Laplace transform \begin{equation}\label{Laplace-mu(t)-INT}\tag{1} f(x)=\int_0^\infty e^{-xt}\textrm{d}\mu(t), \quad x\in(0,\infty), \end{equation} where $\mu(t)$ is non-decreasing and the above integral converges for $x\in(0,\infty)$.
For $\alpha\in\mathbb{R}$, let \begin{equation*} h_\alpha(t)=t^{t[\psi(t)-\ln t]-\alpha}, \quad t>0. \end{equation*} It is easy to prove that the necessary condition for $h_\alpha(t)$ to be logarithmically completely monotonic on $(0,\infty)$ is $\alpha\ge-\frac12$.
Is the necessary condition $\alpha\ge-\frac12$ a sufficient condition for $h_\alpha(t)$ to be logarithmically completely monotonic on $(0,\infty)$?
In the paper [3] below, the function $h_\gamma(t)$ has been proved to be logarithmically completely monotonic on $(0,1)$, where $\gamma=0.577\dotsc$ is the Euler-Mascheroni constant.
A hint: the function $t[\psi(t)-\ln t]+\frac{1}{2}$ is completely monotonic on $(0,\infty)$. See the papers [6, 8] below.
References
- C. Berg, Integral representation of some functions related to the gamma function, Mediterr. J. Math. 1 (2004), no. 4, 433--439; available online at https://doi.org/10.1007/s00009-004-0022-6.
- Bai-Ni Guo and Feng Qi, A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, University Politehnica of Bucharest Scientific Bulletin Series A---Applied Mathematics and Physics 72 (2010), no. 2, 21--30.
- Bai-Ni Guo and Feng Qi, Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions, Global Journal of Mathematical Analysis 3 (2015), no. 2, 77--80; available online at https://doi.org/10.14419/gjma.v3i2.4605.
- 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.
- 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.
- Feng Qi, Bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function, Arab Journal of Basic and Applied Sciences 28 (2021), no. 1, 314--318; available online at https://doi.org/10.1080/25765299.2021.1962060.
- Feng Qi and Chao-Ping Chen, A complete monotonicity property of the gamma function, Journal of Mathematical Analysis and Applications 296 (2004), no. 2, 603--607; available online at https://doi.org/10.1016/j.jmaa.2004.04.026.
- Mansour Mahmoud and Feng Qi, Bounds for completely monotonic degrees of remainders in asymptotic expansions of the digamma function, Mathematical Inequalities & Applications 25 (2022), no. 1, 291--306; available online at https://doi.org/10.7153/mia-2022-25-17.
- R. L. Schilling, R. Song, and Z. Vondracek, Bernstein Functions---Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012; available online at https://doi.org/10.1515/9783110269338.
In the paper "Mohamed Bouali, On some complete monotonic functions, arXiv (2022), 16 pages; available online at http://arxiv.org/abs/2206.01527v1", this problem has been moved forward slightly as follows.
Proposition 1.8. For $\alpha\ge-\frac14$, the function $h_\alpha(t)$ is logarithmically completely monotonic on $(0,\infty)$.
