Proving the complete monotonicity of $f$ given $(−\log f)′$ is completely monotone

Question

A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$.

Prove that if $(−\log f(x))′$ is completely monotonic, then $f(x)$ is also completely monotonic.

There are a few papers that use this without explicitly displaying the proof. It may help to display the function as $g(x)=(−\log f(x))′$ then put $f(x)=\exp(−g(x))$ but I have not yet find the pattern to induct the theorem.

Answer 1

It is not hard to prove the result but the easy proof needs a detour through absolutely monotonic functions ($g \ge 0$ s.t all its derivatives are also positive/non-negative) since while essentially equivalent to completely monotonic functions by $f(x)=g(-x)$, their closure properties are much easier to state and obvious to prove:

1: Let $g(x)=f(-x)$, $g$ defined on $(-\infty,0)$ and to prove $f$ completely monotonic is equivalent to proving $g$ absolutely monotonic

2: Any positive/non-negative primitive of an absolutely monotonic functions is absolutely monotonic trivially from the definition

3: $(−\log f(y))′=(\log g(x))′, y=-x$, hence by the hypothesis and the above, $\log g(x)+C$ is absolutely monotonic for some $C>0$ since $\log g$ is a primitive of an absolutely monotonic function on $(-\infty,0)$, so in particular it is finite at $-\infty$

4: If $h_1,h_2$ are absolutely monotonic, their composition is also absolutely monotonic wherever defined since positive/non-negative numbers form a cone

5; $e^x$ is absolutely monotonic everywhere hence $g(x)=\frac{e^{\log g(x)+C}}{e^C}$ is absolutely monotonic on $(-\infty,0)$, so by point $1$, $f$ is completely monotonic on $(0, \infty)$

Answer 2

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 $n\in\mathbb{N}$.

A positive function $f$ is said to be logarithmically completely monotonic on an interval $I$ if its logarithm $\ln f$ satisfies \begin{equation*} 0\le(-1)^n[\ln f(x)]^{(n)}<\infty \end{equation*} for all $n\in\mathbb{N}$ on $I$.

A logarithmically completely function on an interval $I$ must be also completely monotonic on $I$, but not conversely.

The following references contain the concepts and conclusions mentioned above.

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. 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.
4. 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.