{"paper":{"title":"Further refinements of the Cauchy-Schwarz inequality for matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mojtaba Bakherad","submitted_at":"2014-09-19T07:40:53Z","abstract_excerpt":"Let $A, B$ and $X$ be $n\\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard inequality. In particular, we establish the inequality \\begin{align*} |||\\,|A^{1\\over2}XB^{1\\over2}|^r|||^2&\\leq|||\\,|A^{t}XB^{1-s}|^r||| \\,\\,\\,|||\\,|A^{1-t}XB^{s}|^r|||\\\\& \\leq\\max \\{|||\\,|AX|^r||| \\,\\,\\,|||\\,|XB|^r|||,|||\\,|AXB|^r||| \\,\\,\\,|||\\,|X|^r|||\\}, \\end{align*} where $s,t\\in[0,1]$ and $r\\geq0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5535","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"}