Uniqueness of nontrivially complete monotonicity for a class of functions involving polygamma functions

TL;DR

This paper proves that among a class of functions involving polygamma functions, only one specific function exhibits nontrivial complete monotonicity, highlighting its unique mathematical property.

Contribution

The paper establishes the uniqueness of the nontrivially completely monotonic function within a class involving polygamma functions using two different methods.

Findings

Only f_{1,2}(x) is nontrivially completely monotonic.

Functions f_{1,2}(x) and f_{m,2n-1}(x) are completely monotonic.

Other functions f_{m,2n}(x) are not monotonic or sign-preserving.

Abstract

For $m,n\in\mathbb{N}$, let $f_{m,n}(x)=\bigr[\psi^{(m)}(x)\bigl]^2+\psi^{(n)}(x)$ on $(0,\infty)$. In the present paper, we prove using two methods that, among all $f_{m,n}(x)$ for $m,n\in\mathbb{N}$, only $f_{1,2}(x)$ is nontrivially completely monotonic on $(0,\infty)$. Accurately, the functions $f_{1,2}(x)$ and $f_{m,2n-1}(x)$ are completely monotonic on $(0,\infty)$, but the functions $f_{m,2n}(x)$ for $(m,n)\ne(1,1)$ are not monotonic and does not keep the same sign on $(0,\infty)$.