{"paper":{"title":"Characterizations of Operator Monotonicity via Operator Means and Applications to Operator Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Pattrawut Chansangiam","submitted_at":"2015-06-23T09:30:23Z","abstract_excerpt":"We prove that a continuous function $f:(0,\\infty) \\to (0,\\infty)$ is operator monotone increasing if and only if $f(A \\: !_t \\: B) \\leqs f(A) \\: !_t \\: f(B)$ for any positive operators $A,B$ and scalar $t \\in [0,1]$. Here, $!_t$ denotes the $t$-weighted harmonic mean. As a counterpart, $f$ is operator monotone decreasing if and only if the reverse of preceding inequality holds. Moreover, we obtain many characterizations of operator-monotone increasingness/decreasingness in terms of operator means. These characterizations lead to many operator inequalities involving means."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06922","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"}