{"paper":{"title":"Bounds for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Feng Qi, Wei-Dong Jiang","submitted_at":"2014-02-19T06:03:43Z","abstract_excerpt":"In the paper, the authors find the greatest value $\\lambda$ and the least value $\\mu $ such that the double inequality \\begin{multline*} C(\\lambda a+(1-\\lambda)b,\\lambda b+(1-\\lambda )a)<\\alpha A(a,b)+(1-\\alpha)T(a,b)\\\\ < C(\\mu a+(1-\\mu)b,\\mu b+(1-\\mu )a) \\end{multline*} holds for all $\\alpha\\in(0,1)$ and $a,b>0$ with $a\\ne b$, where $$ C(a,b)=\\frac{a^{2}+b^{2}}{a+b},\\quad A(a,b)=\\frac{a+b}2, $$ and $$ T(a,b)=\\frac{2}{\\pi}\\int_{0}^{{\\pi}/{2}}\\sqrt{a^2\\cos^2\\theta+b^2\\sin^2\\theta}\\,d\\theta$$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.4561","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"}