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

Abstract

For $m,n\in \mathbb{N}$, let $f_{m,n}(x)=[{\psi }^{(m)}(x){]}^2+{\psi }^{(n)}(x)$ 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 $(0,\infty )$.

9 pages

References in corpus (2)

• Complete monotonicity of some functions involving polygamma functions
• Two new proofs of the complete monotonicity of a function involving the psi function

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• Some inequalities for the trigamma function in terms of the digamma function