{"paper":{"title":"On harmonic representation of means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alfred Witkowski","submitted_at":"2013-10-11T09:23:13Z","abstract_excerpt":"We characterize continuous, symmetric and homogeneous means $M$ that can be represented in the form \\begin{equation*} \\frac{1}{M(x,y)}=\\int_0^1 \\frac{dt}{N\\left(\\tfrac{x+y}{2}-t\\tfrac{x-y}{2},\\tfrac{x+y}{2}+t\\tfrac{x-y}{2}\\right)}. \\end{equation*} New inequalities for means are derived from such representation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.3063","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"}