{"paper":{"title":"Inequalities related to Bourin and Heinz means with a complex parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Alejandro Varela, Gabriel Larotonda, Rene Elencwajg, Tamara Bottazzi","submitted_at":"2014-03-28T18:08:27Z","abstract_excerpt":"A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\\le t\\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for $t\\in [1/4;3/4]$. In this paper, using complex methods we extend this result to complex values of the parameter $t=z$ in the strip $\\{z \\in {\\mathbb C}: Re(z) \\in [1/4;3/4]\\}$. We give an elementary proof of the fact that equality holds for some $z$ in the strip if and only if $A$ and $B$ commute. We al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7472","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"}