Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete monotonicity of a family of functions involving the tri- and
  tetra-gamma functions   The psi function $\psi(x)$ is defined by
$\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}$ and $\psi^{(i)}(x)$ for $i\in\mathbb{N}$
denote polygamma functions, where $\Gamma(x)$ is the gamma function. In this
paper, we prove that the function $$ [\psi'(x)]^2+\psi"(x)-\frac{x^2+\lambda
x+12}{12x^4(x+1)^2} $$ is completely monotonic on $(0,\infty)$ if and only if
$\lambda\le0$, and so is its negative if and only if $\lambda\ge4$. From this,
some inequalities are refined and sharpened.

13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th