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For example, for any real number $r \\in \\{1\\} \\cup [2, \\infty)$, positive semi-definite matrices $A_i,\\ B_i,\\ C_i\\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_\\chi, d_\\xi$ such as the determinant and permanent, etc., we have \\begin{eqnarray*}&&\\left(d_\\chi(A_1+B_1+C_1)d_\\xi(A_2+B_2+C_2)\\right)^r \\\\ &&\\hskip 1in\n  + \\left(d_\\chi(A_1)d_\\xi(A_2)\\right)^r\n  + \\left(d_\\chi(B_1)d_\\xi(B_2)\\right)^r + \\left(d_\\chi(C_1)d_\\xi(C_2)\\right)^r \\\\ & \\ge &\\left("},"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":"1605.06984","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-05-23T11:54:08Z","cross_cats_sorted":[],"title_canon_sha256":"2a7a05f8646891d642c11904ca656681789f30b9b37940195e09d878a732ac86","abstract_canon_sha256":"94cee0f279af4cb88aa15431bf443b19691e3a4157968cb75e93bc303ca231f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:40.615392Z","signature_b64":"EdUANYzP4k0WTF+ARriyjFGug5vPGa0wsd7BtQnyF5EOvvXN9236rWoC34UaFhzKmA6Vmc/pNwItVflrtH+PCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e0993b2bd98561ee47ac5bb0605e4024f7019abffe829211ef2c26cc7ad76fb","last_reissued_at":"2026-05-18T01:06:40.614762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:40.614762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Inequalities on generalized matrix functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Chi-Kwong Li, Qing-Wen Wang, Shaowu Huang, Yiu-Tung Poon","submitted_at":"2016-05-23T11:54:08Z","abstract_excerpt":"We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \\in \\{1\\} \\cup [2, \\infty)$, positive semi-definite matrices $A_i,\\ B_i,\\ C_i\\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_\\chi, d_\\xi$ such as the determinant and permanent, etc., we have \\begin{eqnarray*}&&\\left(d_\\chi(A_1+B_1+C_1)d_\\xi(A_2+B_2+C_2)\\right)^r \\\\ &&\\hskip 1in\n  + \\left(d_\\chi(A_1)d_\\xi(A_2)\\right)^r\n  + \\left(d_\\chi(B_1)d_\\xi(B_2)\\right)^r + \\left(d_\\chi(C_1)d_\\xi(C_2)\\right)^r \\\\ & \\ge &\\left("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06984","kind":"arxiv","version":2},"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":"1605.06984","created_at":"2026-05-18T01:06:40.614861+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.06984v2","created_at":"2026-05-18T01:06:40.614861+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06984","created_at":"2026-05-18T01:06:40.614861+00:00"},{"alias_kind":"pith_short_12","alias_value":"LYEZHMV5TBLB","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LYEZHMV5TBLB5ZD2","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LYEZHMV5","created_at":"2026-05-18T12:30:29.479603+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/LYEZHMV5TBLB5ZD2YW5QMBPEAJ","json":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ.json","graph_json":"https://pith.science/api/pith-number/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/graph.json","events_json":"https://pith.science/api/pith-number/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/events.json","paper":"https://pith.science/paper/LYEZHMV5"},"agent_actions":{"view_html":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ","download_json":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ.json","view_paper":"https://pith.science/paper/LYEZHMV5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.06984&json=true","fetch_graph":"https://pith.science/api/pith-number/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/graph.json","fetch_events":"https://pith.science/api/pith-number/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/action/storage_attestation","attest_author":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/action/author_attestation","sign_citation":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/action/citation_signature","submit_replication":"https://pith.science/pith/LYEZHMV5TBLB5ZD2YW5QMBPEAJ/action/replication_record"}},"created_at":"2026-05-18T01:06:40.614861+00:00","updated_at":"2026-05-18T01:06:40.614861+00:00"}