{"paper":{"title":"The best bounds for Toader mean in terms of the centroidal and arithmetic means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Feng Qi, Yun Hua","submitted_at":"2013-03-11T08:34:23Z","abstract_excerpt":"In the paper, the authors discover the best constants $\\alpha_{1}$, $\\alpha_{2}$, $\\beta_{1}$, and $\\beta_{2}$ for the double inequalities $$ \\alpha_{1}\\bar{C}(a,b)+(1-\\alpha_{1}) A(a,b)< T(a,b) <\\beta_{1} \\bar{C}(a,b)+(1-\\beta_{1})A(a,b) $$ and $$ \\frac{\\alpha_{2}}{A(a,b)}+\\frac{1-\\alpha_{2}}{\\bar{C}(a,b)}<\\frac1{T(a,b)} <\\frac{\\beta_{2}}{A(a,b)}+\\frac{1-\\beta_{2}}{\\bar{C}(a,b)} $$ to be valid for all $a,b>0$ with $a\\ne b$, where $$ \\bar{C}(a,b)=\\frac{2(a^{2}+ab+b^{2})}{3(a+b)},\\quad A(a,b)=\\frac{a+b}2, $$ and $$ T(a,b)=\\frac{2}{\\pi}\\int_{0}^{{\\pi}/{2}}\\sqrt{a^2{\\cos^2{\\theta}}+b^2{\\sin^2{\\th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2451","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"}