{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:HCI2VXAK7AEUU2ZL4GJT5D7APF","short_pith_number":"pith:HCI2VXAK","schema_version":"1.0","canonical_sha256":"3891aadc0af8094a6b2be1933e8fe0797ead38cf6f5e146e8aa149b88a3abcba","source":{"kind":"arxiv","id":"1605.03516","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1605.03516","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-11T17:14:03Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"2a17381a7cc0b94ee58fe5f7d875043cc679b0feb1f9b916f194759bbb3e7fbf","abstract_canon_sha256":"b9767e8200076d19a7441ab0243095c558827bd78da876f0d1d48739fc9afd23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:05.231351Z","signature_b64":"IZ6YCzRiX1b1Ac9T5wYWou/ZNYfl06MOa75wB0vTY830EBDgZGr97lXdOjv+zso+/FUDRFYaIN932L/sGjaRDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3891aadc0af8094a6b2be1933e8fe0797ead38cf6f5e146e8aa149b88a3abcba","last_reissued_at":"2026-05-18T01:15:05.230902Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:05.230902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.03516","created_at":"2026-05-18T01:15:05.230964+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.03516v1","created_at":"2026-05-18T01:15:05.230964+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.03516","created_at":"2026-05-18T01:15:05.230964+00:00"},{"alias_kind":"pith_short_12","alias_value":"HCI2VXAK7AEU","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"HCI2VXAK7AEUU2ZL","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"HCI2VXAK","created_at":"2026-05-18T12:30:19.053100+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF","json":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF.json","graph_json":"https://pith.science/api/pith-number/HCI2VXAK7AEUU2ZL4GJT5D7APF/graph.json","events_json":"https://pith.science/api/pith-number/HCI2VXAK7AEUU2ZL4GJT5D7APF/events.json","paper":"https://pith.science/paper/HCI2VXAK"},"agent_actions":{"view_html":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF","download_json":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF.json","view_paper":"https://pith.science/paper/HCI2VXAK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.03516&json=true","fetch_graph":"https://pith.science/api/pith-number/HCI2VXAK7AEUU2ZL4GJT5D7APF/graph.json","fetch_events":"https://pith.science/api/pith-number/HCI2VXAK7AEUU2ZL4GJT5D7APF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF/action/storage_attestation","attest_author":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF/action/author_attestation","sign_citation":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF/action/citation_signature","submit_replication":"https://pith.science/pith/HCI2VXAK7AEUU2ZL4GJT5D7APF/action/replication_record"}},"created_at":"2026-05-18T01:15:05.230964+00:00","updated_at":"2026-05-18T01:15:05.230964+00:00"}