{"paper":{"title":"Some inequalities for the matrix Heron mean","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Dinh Trung Hoa","submitted_at":"2016-05-11T17:14:03Z","abstract_excerpt":"Let $A, B$ be positive definite matrices, $p=1, 2$ and $r\\ge 0$. It is shown that \\begin{equation*} ||A+ B + r(A\\sharp_t B+A\\sharp_{1-t} B)||_p \\le ||A+ B + r(A^{t}B^{1-t} + A^{1-t}B^t)||_p. \\end{equation*} We also prove that for positive definite matrices $A$ and $B$ \\begin{equation*}\\label{det} \\Dt (P_{t}(A, B)) \\le \\Dt (Q_{t}(A, B)), \\end{equation*} where $Q_t(A, B)= \\big(\\frac{A^t+B^t}{2}\\big)^{1/t}$ and $P_t(A, B)$ is the $t$-power mean of $A$ and $B$. As a consequence, we obtain the determinant inequality for the matrix Heron mean: for any positive definite matrices $A$ and $B,$ $$ \\Dt(A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.03516","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"}