{"paper":{"title":"An extension of an inequality for ratios of gamma 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":"2009-02-15T01:52:21Z","abstract_excerpt":"In this paper, we prove that for $x+y>0$ and $y+1>0$ the inequality {equation*} \\frac{[\\Gamma(x+y+1)/\\Gamma(y+1)]^{1/x}}{[\\Gamma(x+y+2)/\\Gamma(y+1)]^{1/(x+1)}} <\\biggl(\\frac{x+y}{x+y+1}\\biggr)^{1/2} {equation*} is valid if $x>1$ and reversed if $x<1$ and that the power $\\frac12$ is the best possible, where $\\Gamma(x)$ is the Euler gamma function. This extends the result in [Y. Yu, \\textit{An inequality for ratios of gamma functions}, J. Math. Anal. Appl. \\textbf{352} (2009), no.~2, 967\\nobreakdash--970.] and resolves an open problem posed in [B.-N. Guo and F. Qi, \\emph{Inequalities and monoton"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.2513","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"}