Does complete monotonicity of f imply log-concavity of f?  Let f be a completely monotonic function with $f(0)=1$, that is, 
$ f:[0, \infty) \rightarrow (0,1] $. My question is:
Is f log concave, that is, is $(logf)''<0$ or equivalently $ f f''< f'^2 $. ?
And what hapens if $f(0)=\infty$, that is if the function is: 
$ f:(0, \infty) \rightarrow (0,\infty) $.
 A: A counterexample in the second case is $f(x) = e^{1/x}$.  A counterexample in the first case is then $f(x) = e^{1/(x+1) - 1}$.
A: Exercise 6 of this book shows that if $f: (0,\infty) \to \mathbb{R}$ is completely monotonic, then it must be log-convex. Hence, your second claim holds, with concavity replaced by convexity.
A: Actually, I meant log-convex instead of log-concave. I missed a minus sign in my derivations and this led to the conjecture that log(f) should be concave. I have found a more complete answer in this paper:
Inequalities for Real Powers of Completely Monotonic Functions
H. van Haeringen
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 
Volume 210, Issue 1, 1 June 1997, Pages 102–113
Theorem 1 establishes a series of inequalities for the derivatives of c.m. functions. In particular, by taking n=0 and m=1 in 3.2 we get the log-convexity of f.
I suggest to read this paper because of the relrevance of Theorem 1.
A: A non-negative function $f(x)$ is said to be completely monotonic on an interval $I$ if $f(x)$ 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 $n\in\mathbb{N}=\{1,2,3,\dotsc\}$.
If a function $f(x)$ is non-identically zero and completely monotonic on $(0,\infty)$, then $f(x)$ and its derivatives $f^{(n)}(x)$ for $n\in\mathbb{N}$ are impossibly equal to $0$ on $(0,\infty)$.
As for completely monotonic functions, there are two kinds of convexities:

*

*If a function $f(x)$ is completely monotonic on an interval $I$, by the above definition, it is trivial that the function $f(x)$ is surely convex, that is, $f''(x)\ge0$, on the interval $I$.

*If a function $f(x)$ is completely monotonic on the infinite interval $(0,\infty)$, then the derivative sequence $f^{(n)}(x)$ in $n\ge0$ for $x\in(0,\infty)$ is surely logarithmically convex in $n\ge0$, that is,
\begin{equation}
\frac{f^{(i)}(x)}{f^{(i+1)}(x)}\ge\frac{f^{(i+1)}(x)}{f^{(i+2)}(x)}, \quad i=0,1,2,\dotsc, \quad x\in(0,\infty).
\end{equation}
A positive function $f(x)$ is said to be logarithmically completely monotonic on an interval $I$ if its logarithm $\ln f(x)$ 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 definition of logarithmically completely monotonic functions and the above relation between completely monotonic functions and logarithmically completely monotonic functions demonstrate that, if a function $f(x)$ is completely monotonic, but not logarithmically completely monotonic, on an interval $I$, then it is possible, but not sure, that the completely monotonic function $f(x)$ is logarithmically concave or logarithmically convex on $I$.
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.

*D. S. Mitrinovic, J. E. Pecaric, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993; available online at https://doi.org/10.1007/978-94-017-1043-5. (Chapter XIII)

*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.

*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.

*Y. Shuang, B.-N. Guo, and F. Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 115 (2021), no. 3, Paper No. 135, 12 pages; available online at https://doi.org/10.1007/s13398-021-01071-x.

*D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946. (Chapter IV)

