{"paper":{"title":"Further refinements of the Heinz inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"C. Conde, M. Singh, M.S. Moslehian, R. Kaur","submitted_at":"2013-01-30T20:10:46Z","abstract_excerpt":"The celebrated Heinz inequality asserts that $ 2|||A^{1/2}XB^{1/2}|||\\leq |||A^{\\nu}XB^{1-\\nu}+A^{1-\\nu}XB^{\\nu}|||\\leq |||AX+XB|||$ for $X \\in \\mathbb{B}(\\mathscr{H})$, $A,B\\in \\+$, every unitarily invariant norm $|||\\cdot|||$ and $\\nu \\in [0,1]$. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function $F(\\nu)=|||A^{\\nu}XB^{1-\\nu}+A^{1-\\nu}XB^{\\nu}|||$, some integration techniques and various refinements of the Hermite--Hadamard inequality. In the setting of matrices we prove that \\begin{eqnarray*} &&\\hspace{-0.5cm}\\left|\\left|\\left|A^{\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.7346","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1301.7346/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}