{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:MPLIHZJLCKSKB3OYEXF54N7MES","short_pith_number":"pith:MPLIHZJL","schema_version":"1.0","canonical_sha256":"63d683e52b12a4a0edd825cbde37ec248e2615b63a1dca22e60862f4f27c4499","source":{"kind":"arxiv","id":"1604.00996","version":1},"attestation_state":"computed","paper":{"title":"Some reversed and refined Callebaut inequalities via Kontorovich constant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mojtaba Bakherad","submitted_at":"2016-04-04T19:28:47Z","abstract_excerpt":"In this paper we employ some operator techniques to establish some refinements and reverses of the Callebaut inequality involving the geometric mean and Hadamard product under some mild conditions. In particular, we show\n  \\begin{align*} K&\\left(\\frac{M^{2t-1}}{m^{2t-1}},2\\right)^{r'} \\sum_{j=1}^n(A_j\\sharp_{s}B_j)\\circ \\sum_{j=1}^n(A_j\\sharp_{1-s}B_j) \\nonumber\\\\&\\,\\,+\\left(\\frac{t-s}{t-1/2}\\right)\\left(\\sum_{j=1}^n(A_j\\sharp_{t}B_j)\\circ \\sum_{j=1}^n(A_j\\sharp_{1-t}B_j) -\\sum_{j=1}^n(A_j\\sharp B_j)\\circ \\sum_{j=1}^n(A_j\\sharp B_j)\\right)\\nonumber \\\\&\\leq \\sum_{j=1}^n(A_j\\sharp_{t}B_j)\\circ \\"},"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":"1604.00996","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-04T19:28:47Z","cross_cats_sorted":[],"title_canon_sha256":"3907a5d5d3e2a1c69cb9040dc13d3e6acb8c25a81e304707967347bb08d7f15a","abstract_canon_sha256":"65c39337d46b2774a50f3ae0a7152f334dd3203cb8cdf3ac6bcbde56b0d94f0a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:47.517888Z","signature_b64":"s9YZLRtSa9cPWOQBvE4NPLA654QkKtxjegK2CPjUeR9NSU4RGjCmzAQOEH7+opJr/JANF/T8ZtYqWAB4EkKzDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63d683e52b12a4a0edd825cbde37ec248e2615b63a1dca22e60862f4f27c4499","last_reissued_at":"2026-05-18T01:17:47.517233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:47.517233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some reversed and refined Callebaut inequalities via Kontorovich constant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mojtaba Bakherad","submitted_at":"2016-04-04T19:28:47Z","abstract_excerpt":"In this paper we employ some operator techniques to establish some refinements and reverses of the Callebaut inequality involving the geometric mean and Hadamard product under some mild conditions. In particular, we show\n  \\begin{align*} K&\\left(\\frac{M^{2t-1}}{m^{2t-1}},2\\right)^{r'} \\sum_{j=1}^n(A_j\\sharp_{s}B_j)\\circ \\sum_{j=1}^n(A_j\\sharp_{1-s}B_j) \\nonumber\\\\&\\,\\,+\\left(\\frac{t-s}{t-1/2}\\right)\\left(\\sum_{j=1}^n(A_j\\sharp_{t}B_j)\\circ \\sum_{j=1}^n(A_j\\sharp_{1-t}B_j) -\\sum_{j=1}^n(A_j\\sharp B_j)\\circ \\sum_{j=1}^n(A_j\\sharp B_j)\\right)\\nonumber \\\\&\\leq \\sum_{j=1}^n(A_j\\sharp_{t}B_j)\\circ \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00996","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":"1604.00996","created_at":"2026-05-18T01:17:47.517328+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.00996v1","created_at":"2026-05-18T01:17:47.517328+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00996","created_at":"2026-05-18T01:17:47.517328+00:00"},{"alias_kind":"pith_short_12","alias_value":"MPLIHZJLCKSK","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MPLIHZJLCKSKB3OY","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MPLIHZJL","created_at":"2026-05-18T12:30:32.724797+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/MPLIHZJLCKSKB3OYEXF54N7MES","json":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES.json","graph_json":"https://pith.science/api/pith-number/MPLIHZJLCKSKB3OYEXF54N7MES/graph.json","events_json":"https://pith.science/api/pith-number/MPLIHZJLCKSKB3OYEXF54N7MES/events.json","paper":"https://pith.science/paper/MPLIHZJL"},"agent_actions":{"view_html":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES","download_json":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES.json","view_paper":"https://pith.science/paper/MPLIHZJL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.00996&json=true","fetch_graph":"https://pith.science/api/pith-number/MPLIHZJLCKSKB3OYEXF54N7MES/graph.json","fetch_events":"https://pith.science/api/pith-number/MPLIHZJLCKSKB3OYEXF54N7MES/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES/action/storage_attestation","attest_author":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES/action/author_attestation","sign_citation":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES/action/citation_signature","submit_replication":"https://pith.science/pith/MPLIHZJLCKSKB3OYEXF54N7MES/action/replication_record"}},"created_at":"2026-05-18T01:17:47.517328+00:00","updated_at":"2026-05-18T01:17:47.517328+00:00"}