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Also, assume that $A \\nabla_\\lambda B:=(1-\\lambda)A+\\lambda B$ and $A \\sharp_\\lambda B:=A^{\\frac{1}{2}}\\left(A^{-\\frac{1}{2}}BA^{-\\frac{1}{2}}\\right)^\\lambda A^{\\frac{1}{2}}$ are arithmetic and geometric means of $A, B$, respectively, where $0 < \\lambda < 1$. We show that if $A$ and $B$ are commuting, then $$ B'~\\nabla_\\lambda~A' - B'~\\sharp_\\lambda~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":"1806.10806","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-28T07:42:37Z","cross_cats_sorted":[],"title_canon_sha256":"12773d9b6893ea7ca75cbccbd4a985c92c232bd88e722d13b1574bebbd1ba718","abstract_canon_sha256":"473f9083eb866d95cccce5641021faf9dfca14c8a670d83d9ef77d9bf652a0ad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:07.873396Z","signature_b64":"I13Ggqmkcvpyz5RcIJxljWDYRSmcWMSXmxvxGd0yv02OUsToAxFmMpfF66kIl0hxcESKg2U0qbn4IWw585tQAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3bae395417e7b17e4634c448aa14ff3834b623f5aaee925b5a596ed51291615a","last_reissued_at":"2026-05-18T00:12:07.872736Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:07.872736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Alzer Inequality for Hilbert Spaces Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ali Morassaei, Farzollah Mirzapour","submitted_at":"2018-06-28T07:42:37Z","abstract_excerpt":"In this paper, we give the Alzer inequality for Hilbert space operators as follows:\n  Let $A, B$ be two selfadjoint operators on a Hilbert space $\\mathcal H$ such that $0 < A, B \\le \\frac{1}{2}I$, where $I$ is identity operator on $\\mathcal H$. Also, assume that $A \\nabla_\\lambda B:=(1-\\lambda)A+\\lambda B$ and $A \\sharp_\\lambda B:=A^{\\frac{1}{2}}\\left(A^{-\\frac{1}{2}}BA^{-\\frac{1}{2}}\\right)^\\lambda A^{\\frac{1}{2}}$ are arithmetic and geometric means of $A, B$, respectively, where $0 < \\lambda < 1$. We show that if $A$ and $B$ are commuting, then $$ B'~\\nabla_\\lambda~A' - B'~\\sharp_\\lambda~A' "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.10806","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":"1806.10806","created_at":"2026-05-18T00:12:07.872836+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.10806v1","created_at":"2026-05-18T00:12:07.872836+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.10806","created_at":"2026-05-18T00:12:07.872836+00:00"},{"alias_kind":"pith_short_12","alias_value":"HOXDSVAX46YX","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"HOXDSVAX46YX4RRU","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"HOXDSVAX","created_at":"2026-05-18T12:32:28.185984+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/HOXDSVAX46YX4RRUYREKUFH7HA","json":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA.json","graph_json":"https://pith.science/api/pith-number/HOXDSVAX46YX4RRUYREKUFH7HA/graph.json","events_json":"https://pith.science/api/pith-number/HOXDSVAX46YX4RRUYREKUFH7HA/events.json","paper":"https://pith.science/paper/HOXDSVAX"},"agent_actions":{"view_html":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA","download_json":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA.json","view_paper":"https://pith.science/paper/HOXDSVAX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.10806&json=true","fetch_graph":"https://pith.science/api/pith-number/HOXDSVAX46YX4RRUYREKUFH7HA/graph.json","fetch_events":"https://pith.science/api/pith-number/HOXDSVAX46YX4RRUYREKUFH7HA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA/action/storage_attestation","attest_author":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA/action/author_attestation","sign_citation":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA/action/citation_signature","submit_replication":"https://pith.science/pith/HOXDSVAX46YX4RRUYREKUFH7HA/action/replication_record"}},"created_at":"2026-05-18T00:12:07.872836+00:00","updated_at":"2026-05-18T00:12:07.872836+00:00"}