Mathematics – Classical Analysis and ODEs
Scientific paper
2009-04-07
B.-N. Guo, F. Qi and H. M. Srivastava, Some uniqueness results for the non-trivially complete monotonicity of a class of funct
Mathematics
Classical Analysis and ODEs
9 pages
Scientific paper
10.1080/10652461003748112
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 $(0,\infty)$.
Guo Bai-Ni
Qi Feng
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