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In particular, we establish the inequality \\begin{align*} |||\\,|A^{1\\over2}XB^{1\\over2}|^r|||^2&\\leq|||\\,|A^{t}XB^{1-s}|^r||| \\,\\,\\,|||\\,|A^{1-t}XB^{s}|^r|||\\\\& \\leq\\max \\{|||\\,|AX|^r||| \\,\\,\\,|||\\,|XB|^r|||,|||\\,|AXB|^r||| \\,\\,\\,|||\\,|X|^r|||\\}, \\end{align*} where $s,t\\in[0,1]$ and $r\\geq0$."},"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":"1409.5535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-09-19T07:40:53Z","cross_cats_sorted":[],"title_canon_sha256":"4794df3dd985c60818c5abcfe684df291314f2d9079ea13e80fe07a185b8b0f5","abstract_canon_sha256":"87534a5b360d529533337d7c4e29556c41dd27a4ede77fea7c4f86db2c4789db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:33:00.950607Z","signature_b64":"yqS1uSuY8pNdaXRGJ6LaKQIVnFAt7hj/t5LM5iURkXvKc/XpZnBdSk12af+ERlgYmMBQ9+Z1rjDoNv7WK07vAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53dbee57bc0224035388c29d9b3d706b44786b6c9b83dc1a564acdc2c7e345a9","last_reissued_at":"2026-05-18T02:33:00.950273Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:33:00.950273Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Further refinements of the Cauchy-Schwarz inequality for matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mojtaba Bakherad","submitted_at":"2014-09-19T07:40:53Z","abstract_excerpt":"Let $A, B$ and $X$ be $n\\times n$ matrices such that $A, B$ are positive semidefinite. We present some refinements of the matrix Cauchy-Schwarz inequality by using some integration techniques and various refinements of the Hermite--Hadamard inequality. In particular, we establish the inequality \\begin{align*} |||\\,|A^{1\\over2}XB^{1\\over2}|^r|||^2&\\leq|||\\,|A^{t}XB^{1-s}|^r||| \\,\\,\\,|||\\,|A^{1-t}XB^{s}|^r|||\\\\& \\leq\\max \\{|||\\,|AX|^r||| \\,\\,\\,|||\\,|XB|^r|||,|||\\,|AXB|^r||| \\,\\,\\,|||\\,|X|^r|||\\}, \\end{align*} where $s,t\\in[0,1]$ and $r\\geq0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5535","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":"1409.5535","created_at":"2026-05-18T02:33:00.950330+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.5535v1","created_at":"2026-05-18T02:33:00.950330+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.5535","created_at":"2026-05-18T02:33:00.950330+00:00"},{"alias_kind":"pith_short_12","alias_value":"KPN64V54AISA","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KPN64V54AISAGU4I","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KPN64V54","created_at":"2026-05-18T12:28:35.611951+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/KPN64V54AISAGU4IYKOZWPLQNN","json":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN.json","graph_json":"https://pith.science/api/pith-number/KPN64V54AISAGU4IYKOZWPLQNN/graph.json","events_json":"https://pith.science/api/pith-number/KPN64V54AISAGU4IYKOZWPLQNN/events.json","paper":"https://pith.science/paper/KPN64V54"},"agent_actions":{"view_html":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN","download_json":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN.json","view_paper":"https://pith.science/paper/KPN64V54","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.5535&json=true","fetch_graph":"https://pith.science/api/pith-number/KPN64V54AISAGU4IYKOZWPLQNN/graph.json","fetch_events":"https://pith.science/api/pith-number/KPN64V54AISAGU4IYKOZWPLQNN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN/action/storage_attestation","attest_author":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN/action/author_attestation","sign_citation":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN/action/citation_signature","submit_replication":"https://pith.science/pith/KPN64V54AISAGU4IYKOZWPLQNN/action/replication_record"}},"created_at":"2026-05-18T02:33:00.950330+00:00","updated_at":"2026-05-18T02:33:00.950330+00:00"}