{"paper":{"title":"Uniqueness of nontrivially complete monotonicity for a class of functions involving polygamma functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bai-Ni Guo, Feng Qi","submitted_at":"2009-04-07T10:34:01Z","abstract_excerpt":"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 $(0,\\infty)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.1104","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}