Complete monotonicity of a family of functions involving the tri- and tetra-gamma functions

Abstract

The psi function $\psi (x)$ is defined by $\psi (x)=\frac{{\Gamma }^{\prime }(x)}{\Gamma (x)}$ 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 $$[{\psi }^{\prime }(x){]}^2+\psi "(x)-\frac{x^2+\lambda x+12}{12x^4(x+1{)}^2}$$ is completely monotonic on $(0,\infty )$ if and only if $\lambda \le 0$, and so is its negative if and only if $\lambda \ge 4$. From this, some inequalities are refined and sharpened.

arXiv comment11 pages