# Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

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  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

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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 $$(0,\infty)$$.

• Big thanks for all references !
– Dens
Nov 10, 2021 at 20:04
• @Dens This is academic standard and my own style. Welcome. Nov 12, 2021 at 0:56
• Would you care to check out my question regarding complete monotonicity math.stackexchange.com/q/4767589/64809?
– Hans
Sep 12 at 15:56