A completely monotonic function 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 the polygamma functions, where $\Gamma (x)$ is the gamma function. In this paper we prove that a function involving the difference between $[{\psi }^{\prime }(x){]}^2+{\psi }^{\prime \prime }(x)$ and a proper fraction of $x$ is completely monotonic on $(0,\infty )$.

10 pages

References in corpus (3)

• A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications
• Uniqueness of nontrivially complete monotonicity for a class of functions involving polygamma functions
• Completely monotonic degree of a function involving the tri- and tetra-gamma functions

Cited by in corpus (5)

• Bounds for the ratio of two gamma functions--From Wendel's limit to Elezović-Giordano-Pečarić's theorem
• A logarithmically completely monotonic function involving the ratio of gamma functions
• Completely monotonic degree of a function involving the tri- and tetra-gamma functions
• Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic
• Some inequalities for the trigamma function in terms of the digamma function