{"paper":{"title":"A double inequality for bounding Toader mean by the centroidal mean","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Feng Qi, Yun Hua","submitted_at":"2014-02-20T14:56:30Z","abstract_excerpt":"In the paper, the authors find the best numbers $\\alpha$ and $\\beta$ such that $$ \\overline{C}\\bigl(\\alpha a+(1-\\alpha)b,\\alpha b+(1-\\alpha)a\\bigr)<T(a,b) <\\overline{C}\\bigl(\\beta a+(1-\\beta)b,\\beta b+(1-\\beta)a\\bigr) $$ for all $a,b>0$ with $a\\ne b$, where $\\overline{C}(a,b)={2\\bigl(a^2+ab+b^2\\bigr)}{3(a+b)}$ and $T(a,b)=\\frac{2}{\\pi}\\int_{0}^{{\\pi}/{2}}\\sqrt{a^2{\\cos^2{\\theta}}+b^2{\\sin^2{\\theta}}}\\,d\\theta$ denote respectively the centroidal mean and Toader mean of two positive numbers $a$ and $b$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5020","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"}