{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MPLIHZJLCKSKB3OYEXF54N7MES","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"65c39337d46b2774a50f3ae0a7152f334dd3203cb8cdf3ac6bcbde56b0d94f0a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-04T19:28:47Z","title_canon_sha256":"3907a5d5d3e2a1c69cb9040dc13d3e6acb8c25a81e304707967347bb08d7f15a"},"schema_version":"1.0","source":{"id":"1604.00996","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.00996","created_at":"2026-05-18T01:17:47Z"},{"alias_kind":"arxiv_version","alias_value":"1604.00996v1","created_at":"2026-05-18T01:17:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00996","created_at":"2026-05-18T01:17:47Z"},{"alias_kind":"pith_short_12","alias_value":"MPLIHZJLCKSK","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MPLIHZJLCKSKB3OY","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MPLIHZJL","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:b851228000989635a12c9cb8a2eae364393cee479baa69b40e1a894c20b28356","target":"graph","created_at":"2026-05-18T01:17:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 \\","authors_text":"Mojtaba Bakherad","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-04T19:28:47Z","title":"Some reversed and refined Callebaut inequalities via Kontorovich constant"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00996","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:700ed3ae50a4ed5eeda6bc5ba67c82995e1a204a4aa206aafd2b4f4ee7a11972","target":"record","created_at":"2026-05-18T01:17:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"65c39337d46b2774a50f3ae0a7152f334dd3203cb8cdf3ac6bcbde56b0d94f0a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-04T19:28:47Z","title_canon_sha256":"3907a5d5d3e2a1c69cb9040dc13d3e6acb8c25a81e304707967347bb08d7f15a"},"schema_version":"1.0","source":{"id":"1604.00996","kind":"arxiv","version":1}},"canonical_sha256":"63d683e52b12a4a0edd825cbde37ec248e2615b63a1dca22e60862f4f27c4499","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63d683e52b12a4a0edd825cbde37ec248e2615b63a1dca22e60862f4f27c4499","first_computed_at":"2026-05-18T01:17:47.517233Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:47.517233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s9YZLRtSa9cPWOQBvE4NPLA654QkKtxjegK2CPjUeR9NSU4RGjCmzAQOEH7+opJr/JANF/T8ZtYqWAB4EkKzDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:47.517888Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.00996","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:700ed3ae50a4ed5eeda6bc5ba67c82995e1a204a4aa206aafd2b4f4ee7a11972","sha256:b851228000989635a12c9cb8a2eae364393cee479baa69b40e1a894c20b28356"],"state_sha256":"0c0bc1b5830fa717b7a6081b2286857911fdeb744af244057510e984d7f7695b"}