{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:2L6S36W27HYN3AMVUU6VQ4WILR","short_pith_number":"pith:2L6S36W2","canonical_record":{"source":{"id":"1403.7472","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-03-28T18:08:27Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"3407c85439e7ff8ca2ce0260cfc7f8aa9df9dcaaa337ffbc13d24f5b6cfa1cf2","abstract_canon_sha256":"1465af0386a7e4dda7aa78b6a5bee55b43103742788e40a0fe5486868c6c72f1"},"schema_version":"1.0"},"canonical_sha256":"d2fd2dfadaf9f0dd8195a53d5872c85c46f6e8c14a8a086435ace8633b20e412","source":{"kind":"arxiv","id":"1403.7472","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.7472","created_at":"2026-05-18T02:55:19Z"},{"alias_kind":"arxiv_version","alias_value":"1403.7472v1","created_at":"2026-05-18T02:55:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.7472","created_at":"2026-05-18T02:55:19Z"},{"alias_kind":"pith_short_12","alias_value":"2L6S36W27HYN","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"2L6S36W27HYN3AMV","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"2L6S36W2","created_at":"2026-05-18T12:28:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:2L6S36W27HYN3AMVUU6VQ4WILR","target":"record","payload":{"canonical_record":{"source":{"id":"1403.7472","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-03-28T18:08:27Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"3407c85439e7ff8ca2ce0260cfc7f8aa9df9dcaaa337ffbc13d24f5b6cfa1cf2","abstract_canon_sha256":"1465af0386a7e4dda7aa78b6a5bee55b43103742788e40a0fe5486868c6c72f1"},"schema_version":"1.0"},"canonical_sha256":"d2fd2dfadaf9f0dd8195a53d5872c85c46f6e8c14a8a086435ace8633b20e412","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:19.579354Z","signature_b64":"Mq754OzkL+3BFlDxWGRp62559McSJseLYaQ05P9lCCRhHa9axQsQT3ZI/fOzGkmAOzpcxsyZFQVxeQisNQC1AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2fd2dfadaf9f0dd8195a53d5872c85c46f6e8c14a8a086435ace8633b20e412","last_reissued_at":"2026-05-18T02:55:19.578759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:19.578759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1403.7472","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SPWoe0AzMGX7SEXvnXDMZF6Od1JxhcjnEc5N1ZSH3CdACNbS71uiDjE7zkSOqsdR5wg0P3GmufiVS3M1z8+eDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T20:21:45.081157Z"},"content_sha256":"2ed19a611c2b99d94039241d86dbb144482adaface24e743b232574c4f5ffa7a","schema_version":"1.0","event_id":"sha256:2ed19a611c2b99d94039241d86dbb144482adaface24e743b232574c4f5ffa7a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:2L6S36W27HYN3AMVUU6VQ4WILR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Inequalities related to Bourin and Heinz means with a complex parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Alejandro Varela, Gabriel Larotonda, Rene Elencwajg, Tamara Bottazzi","submitted_at":"2014-03-28T18:08:27Z","abstract_excerpt":"A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\\le t\\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for $t\\in [1/4;3/4]$. In this paper, using complex methods we extend this result to complex values of the parameter $t=z$ in the strip $\\{z \\in {\\mathbb C}: Re(z) \\in [1/4;3/4]\\}$. We give an elementary proof of the fact that equality holds for some $z$ in the strip if and only if $A$ and $B$ commute. We al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7472","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:55:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tC46Qs00rvSlF29b3zNXwaMsgpzuEZfOkF93Dd/tNd5BFdWrOwYbbbEmMRgk/W61BB5qkp4tyjVfTUGebFUtAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T20:21:45.081518Z"},"content_sha256":"1ca4a7a18a5e4e1222c09829d925f95406b883633b520886a4f58cc56f103160","schema_version":"1.0","event_id":"sha256:1ca4a7a18a5e4e1222c09829d925f95406b883633b520886a4f58cc56f103160"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2L6S36W27HYN3AMVUU6VQ4WILR/bundle.json","state_url":"https://pith.science/pith/2L6S36W27HYN3AMVUU6VQ4WILR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2L6S36W27HYN3AMVUU6VQ4WILR/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T20:21:45Z","links":{"resolver":"https://pith.science/pith/2L6S36W27HYN3AMVUU6VQ4WILR","bundle":"https://pith.science/pith/2L6S36W27HYN3AMVUU6VQ4WILR/bundle.json","state":"https://pith.science/pith/2L6S36W27HYN3AMVUU6VQ4WILR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2L6S36W27HYN3AMVUU6VQ4WILR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:2L6S36W27HYN3AMVUU6VQ4WILR","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":"1465af0386a7e4dda7aa78b6a5bee55b43103742788e40a0fe5486868c6c72f1","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-03-28T18:08:27Z","title_canon_sha256":"3407c85439e7ff8ca2ce0260cfc7f8aa9df9dcaaa337ffbc13d24f5b6cfa1cf2"},"schema_version":"1.0","source":{"id":"1403.7472","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.7472","created_at":"2026-05-18T02:55:19Z"},{"alias_kind":"arxiv_version","alias_value":"1403.7472v1","created_at":"2026-05-18T02:55:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.7472","created_at":"2026-05-18T02:55:19Z"},{"alias_kind":"pith_short_12","alias_value":"2L6S36W27HYN","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"2L6S36W27HYN3AMV","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"2L6S36W2","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:1ca4a7a18a5e4e1222c09829d925f95406b883633b520886a4f58cc56f103160","target":"graph","created_at":"2026-05-18T02:55:19Z","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":"A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\\le t\\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for $t\\in [1/4;3/4]$. In this paper, using complex methods we extend this result to complex values of the parameter $t=z$ in the strip $\\{z \\in {\\mathbb C}: Re(z) \\in [1/4;3/4]\\}$. We give an elementary proof of the fact that equality holds for some $z$ in the strip if and only if $A$ and $B$ commute. We al","authors_text":"Alejandro Varela, Gabriel Larotonda, Rene Elencwajg, Tamara Bottazzi","cross_cats":["math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-03-28T18:08:27Z","title":"Inequalities related to Bourin and Heinz means with a complex parameter"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7472","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":"sha256:2ed19a611c2b99d94039241d86dbb144482adaface24e743b232574c4f5ffa7a","target":"record","created_at":"2026-05-18T02:55:19Z","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":"1465af0386a7e4dda7aa78b6a5bee55b43103742788e40a0fe5486868c6c72f1","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-03-28T18:08:27Z","title_canon_sha256":"3407c85439e7ff8ca2ce0260cfc7f8aa9df9dcaaa337ffbc13d24f5b6cfa1cf2"},"schema_version":"1.0","source":{"id":"1403.7472","kind":"arxiv","version":1}},"canonical_sha256":"d2fd2dfadaf9f0dd8195a53d5872c85c46f6e8c14a8a086435ace8633b20e412","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d2fd2dfadaf9f0dd8195a53d5872c85c46f6e8c14a8a086435ace8633b20e412","first_computed_at":"2026-05-18T02:55:19.578759Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:19.578759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Mq754OzkL+3BFlDxWGRp62559McSJseLYaQ05P9lCCRhHa9axQsQT3ZI/fOzGkmAOzpcxsyZFQVxeQisNQC1AA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:19.579354Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.7472","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ed19a611c2b99d94039241d86dbb144482adaface24e743b232574c4f5ffa7a","sha256:1ca4a7a18a5e4e1222c09829d925f95406b883633b520886a4f58cc56f103160"],"state_sha256":"02e365e9708e3b129be985ff6487481f04e55a14526dc5c61372cd43fd32d95f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qhI/+IGM4dx/COMY93F6Quz3AcO7ikeVdKs6hebjgLaKScAn5SaDupmpMPI1HzGZZ2aAR+yIgKjQzfMq4trjCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T20:21:45.083474Z","bundle_sha256":"eccb7a352c81d9537a8b910bd499d1ce70322e67c2d2da0878ad9c42546d792c"}}