%0 Journal Article
%T Uniqueness of nontrivially complete monotonicity for a class of functions involving polygamma functions
%A Feng Qi
%A Bai-Ni Guo
%J Mathematics
%D 2009
%I arXiv
%R 10.1080/10652461003748112
%X 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 $(0,\infty)$.
%U http://arxiv.org/abs/0904.1104v2