{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:GR56LI6P25PNLY4J727QUBBCRA","short_pith_number":"pith:GR56LI6P","schema_version":"1.0","canonical_sha256":"347be5a3cfd75ed5e389febf0a042288041ee37a8b7fbb1c7aea8e4922bf0f13","source":{"kind":"arxiv","id":"1301.6848","version":1},"attestation_state":"computed","paper":{"title":"The geometric mean is a Bernstein function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.CA","authors_text":"Feng Qi, Wen-Hui Li, Xiao-Jing Zhang","submitted_at":"2013-01-29T07:23:30Z","abstract_excerpt":"In the paper, the authors establish, by using Cauchy integral formula in the theory of complex functions, an integral representation for the geometric mean of $n$ positive numbers. From this integral representation, the geometric mean is proved to be a Bernstein function and a new proof of the well known AG inequality is provided."},"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":"1301.6848","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-01-29T07:23:30Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"63e557fdf58a3572368beedcb7a494e110a27c40e638a6f7b5e8e341a430ab2a","abstract_canon_sha256":"e2283de3d68fa6de1d8f2dd1f5c4a29f946e9b79756cbd7ee7d7f31828a7483d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:00.043564Z","signature_b64":"6hasmQHhYz55qpM0CMy8vppNgk8CM/DKLyHMApmAH62PzHgiA9ajOCk/SPshR0sLzTECWe9A5hUkqJojnyJZDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"347be5a3cfd75ed5e389febf0a042288041ee37a8b7fbb1c7aea8e4922bf0f13","last_reissued_at":"2026-05-18T02:57:00.043135Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:00.043135Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The geometric mean is a Bernstein function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.CA","authors_text":"Feng Qi, Wen-Hui Li, Xiao-Jing Zhang","submitted_at":"2013-01-29T07:23:30Z","abstract_excerpt":"In the paper, the authors establish, by using Cauchy integral formula in the theory of complex functions, an integral representation for the geometric mean of $n$ positive numbers. From this integral representation, the geometric mean is proved to be a Bernstein function and a new proof of the well known AG inequality is provided."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6848","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":"1301.6848","created_at":"2026-05-18T02:57:00.043193+00:00"},{"alias_kind":"arxiv_version","alias_value":"1301.6848v1","created_at":"2026-05-18T02:57:00.043193+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.6848","created_at":"2026-05-18T02:57:00.043193+00:00"},{"alias_kind":"pith_short_12","alias_value":"GR56LI6P25PN","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"GR56LI6P25PNLY4J","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"GR56LI6P","created_at":"2026-05-18T12:27:45.050594+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/GR56LI6P25PNLY4J727QUBBCRA","json":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA.json","graph_json":"https://pith.science/api/pith-number/GR56LI6P25PNLY4J727QUBBCRA/graph.json","events_json":"https://pith.science/api/pith-number/GR56LI6P25PNLY4J727QUBBCRA/events.json","paper":"https://pith.science/paper/GR56LI6P"},"agent_actions":{"view_html":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA","download_json":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA.json","view_paper":"https://pith.science/paper/GR56LI6P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1301.6848&json=true","fetch_graph":"https://pith.science/api/pith-number/GR56LI6P25PNLY4J727QUBBCRA/graph.json","fetch_events":"https://pith.science/api/pith-number/GR56LI6P25PNLY4J727QUBBCRA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA/action/storage_attestation","attest_author":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA/action/author_attestation","sign_citation":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA/action/citation_signature","submit_replication":"https://pith.science/pith/GR56LI6P25PNLY4J727QUBBCRA/action/replication_record"}},"created_at":"2026-05-18T02:57:00.043193+00:00","updated_at":"2026-05-18T02:57:00.043193+00:00"}