{"paper":{"title":"Monotonicity and log-behavior of some functions related to the Euler Gamma function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bao-Xuan Zhu","submitted_at":"2013-09-23T04:37:19Z","abstract_excerpt":"The aim of this paper is to develop analytic techniques to deal with certain monotonicity of combinatorial sequences. (1) A criterion for the monotonicity of the function $\\sqrt[x]{f(x)}$ is given, which is a continuous analog for one result of Wang and Zhu. (2) The log-behavior of the functions $\\theta(x)=\\sqrt[x]{2 \\zeta(x)\\Gamma(x+1)}$ and $F(x)=\\sqrt[x]{\\frac{\\Gamma(ax+b+1)}{\\Gamma(c x+d+1)\\Gamma(e x+f+1)}}$ is considered, where $\\zeta(x)$ and $\\Gamma(x)$ are the Riemann zeta function and the Euler Gamma function, respectively. As consequences, the strict log-concavities of the function $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5693","kind":"arxiv","version":3},"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"}