{"paper":{"title":"Alzer Inequality for Hilbert Spaces Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ali Morassaei, Farzollah Mirzapour","submitted_at":"2018-06-28T07:42:37Z","abstract_excerpt":"In this paper, we give the Alzer inequality for Hilbert space operators as follows:\n  Let $A, B$ be two selfadjoint operators on a Hilbert space $\\mathcal H$ such that $0 < A, B \\le \\frac{1}{2}I$, where $I$ is identity operator on $\\mathcal H$. Also, assume that $A \\nabla_\\lambda B:=(1-\\lambda)A+\\lambda B$ and $A \\sharp_\\lambda B:=A^{\\frac{1}{2}}\\left(A^{-\\frac{1}{2}}BA^{-\\frac{1}{2}}\\right)^\\lambda A^{\\frac{1}{2}}$ are arithmetic and geometric means of $A, B$, respectively, where $0 < \\lambda < 1$. We show that if $A$ and $B$ are commuting, then $$ B'~\\nabla_\\lambda~A' - B'~\\sharp_\\lambda~A' "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10806","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"}