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We show that for any operator mean $\\sigma$ with the representing function $f$, the double inequality $$ \\omega^{1-\\alpha}(\\Phi(A)#_{\\alpha}\\Phi(B))\\le (\\omega\\Phi(A))\\nabla_{\\alpha}\\Phi(B)\\leq \\frac{\\alpha}{\\mu}\\Phi(A\\sigma B) $$ holds, where $\\mu=\\frac{a_{1}b_{1}(f(b_{2}a_{1}^{-1})-f(a_{2}b_{1}^{-1}))}{b_{1}b_{2}-a_{1}a_{2}}, $ $\\nu=\\frac{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":"1204.5049","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-23T12:59:27Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"1dd3174115fa5a7a3c7eb2b504a6b539f2a6610a9d854f75a97650ef79426db8","abstract_canon_sha256":"c1dcdf6334363520971928eca928cd8760499af23d89bd553686baf82bdb1d40"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:23.016301Z","signature_b64":"Nzy+9Oqf884ZVmOf1gJMyfKCqxFNlQAKpIoeEDUNsxCrfgN80eqNLsgzBWVGBRC0+XrdR0eioU6sSTr1mQWdBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"410b28cbc92b12d9f92f2f21fba33731270920307022879ad5ec867a9bf097aa","last_reissued_at":"2026-05-18T03:55:23.015486Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:23.015486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A General Double Inequality Related to Operator Means and Positive Linear Maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"J. S. Aujla, M. Singh, M. S. Moslehian, R. Kaur","submitted_at":"2012-04-23T12:59:27Z","abstract_excerpt":"Let $A,B\\in \\mathbb{B}(\\mathscr{H})$ be such that $0<b_{1}I \\leq A \\leq a_{1}I$ and $0<b_{2}I \\leq B \\leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\\;\\; i=1,2$ and $\\Phi:\\mathbb{B}(\\mathscr{H})\\rightarrow\\mathbb{B}(\\mathscr{K})$ be a positive linear map. We show that for any operator mean $\\sigma$ with the representing function $f$, the double inequality $$ \\omega^{1-\\alpha}(\\Phi(A)#_{\\alpha}\\Phi(B))\\le (\\omega\\Phi(A))\\nabla_{\\alpha}\\Phi(B)\\leq \\frac{\\alpha}{\\mu}\\Phi(A\\sigma B) $$ holds, where $\\mu=\\frac{a_{1}b_{1}(f(b_{2}a_{1}^{-1})-f(a_{2}b_{1}^{-1}))}{b_{1}b_{2}-a_{1}a_{2}}, $ $\\nu=\\frac{a_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.5049","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":"1204.5049","created_at":"2026-05-18T03:55:23.015628+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.5049v1","created_at":"2026-05-18T03:55:23.015628+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.5049","created_at":"2026-05-18T03:55:23.015628+00:00"},{"alias_kind":"pith_short_12","alias_value":"IEFSRS6JFMJN","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_16","alias_value":"IEFSRS6JFMJNT6JP","created_at":"2026-05-18T12:27:09.501522+00:00"},{"alias_kind":"pith_short_8","alias_value":"IEFSRS6J","created_at":"2026-05-18T12:27:09.501522+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/IEFSRS6JFMJNT6JPF4Q7XIZXGE","json":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE.json","graph_json":"https://pith.science/api/pith-number/IEFSRS6JFMJNT6JPF4Q7XIZXGE/graph.json","events_json":"https://pith.science/api/pith-number/IEFSRS6JFMJNT6JPF4Q7XIZXGE/events.json","paper":"https://pith.science/paper/IEFSRS6J"},"agent_actions":{"view_html":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE","download_json":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE.json","view_paper":"https://pith.science/paper/IEFSRS6J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.5049&json=true","fetch_graph":"https://pith.science/api/pith-number/IEFSRS6JFMJNT6JPF4Q7XIZXGE/graph.json","fetch_events":"https://pith.science/api/pith-number/IEFSRS6JFMJNT6JPF4Q7XIZXGE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE/action/storage_attestation","attest_author":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE/action/author_attestation","sign_citation":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE/action/citation_signature","submit_replication":"https://pith.science/pith/IEFSRS6JFMJNT6JPF4Q7XIZXGE/action/replication_record"}},"created_at":"2026-05-18T03:55:23.015628+00:00","updated_at":"2026-05-18T03:55:23.015628+00:00"}