Is the reciprocal of the inverse tangent function, $\frac1{\arctan x}$, a (logarithmically) completely monotonic function on $(0,\infty)$?

Question

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 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 $I$.

If a function $f$ is non-identically zero and completely monotonic on $(0,\infty)$, then $f$ and its derivatives $f^{(n)}(x)$ for $n\in\mathbb{N}$ are impossibly equal to $0$ on $(0,\infty)$.

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

The Bernstein--Widder theorem characterizes that a necessary and sufficient condition for $f$ to be completely monotonic on $(0,\infty)$ is that \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 $x\in(0,\infty)$.

• Question 1: Is the reciprocal of the inverse tangent function, $\dfrac1{\arctan x}$, a (logarithmically) completely monotonic function on $(0,\infty)$?
• Question 2: If $\dfrac1{\arctan x}$ is a (logarithmically) completely monotonic function on $(0,\infty)$, can one give an explicit expression of the measure $\mu(t)$ in the integral representation \eqref{Laplace-mu(t)-INT} for $f(x)=\dfrac1{\arctan x}$?

The following references contain the concepts and the questions stated 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 Ravi P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, Journal of Inequalities and Applications 2019, Paper No. 36, 42 pages; available online at https://doi.org/10.1186/s13660-019-1976-z.
4. 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.
5. 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.