{"paper":{"title":"Sharp inequalities for 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":"2009-03-11T14:36:24Z","abstract_excerpt":"The main aim of this paper is to prove that the double inequality \\frac{(k-1)!}{\\Bigl\\{x+\\Bigl[\\frac{(k-1)!}{|\\psi^{(k)}(1)|}\\Bigr]^{1/k}\\Bigr\\}^k} +\\frac{k!}{x^{k+1}}<\\bigl|\\psi^{(k)}(x)\\bigr|<\\frac{(k-1)!}{\\bigl(x+\\frac12\\bigr)^k}+\\frac{k!}{x^{k+1}} holds for $x>0$ and $k\\in\\mathbb{N}$ and that the constants $\\Bigl[\\frac{(k-1)!}{|\\psi^{(k)}(1)|}\\Bigr]^{1/k}$ and $\\frac12$ are the best possible. In passing, some related inequalities and (logarithmically) complete monotonicity results concerning the gamma, psi and polygamma functions are surveyed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.1984","kind":"arxiv","version":1},"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"}