{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:JPOA5TST6QAOP6MNMMWXU6DNEX","short_pith_number":"pith:JPOA5TST","schema_version":"1.0","canonical_sha256":"4bdc0ece53f400e7f98d632d7a786d25ccd4cf11ace6dd7dcb3d922b72a6a05f","source":{"kind":"arxiv","id":"1511.04642","version":1},"attestation_state":"computed","paper":{"title":"Landau's theorems for certain biharmonic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Ming-Sheng Liu, Yu-Can Zhu, Zhi-Wen Liu","submitted_at":"2015-11-15T02:13:03Z","abstract_excerpt":"Let $f(z)=h(z)+\\overline{g(z)}$ be a harmonic mapping of the unit disk $U$. In this paper, the sharp coefficient estimates for bounded planar harmonic mappings are established, the sharp coefficient estimates for normalized planar harmonic mappings with $|h(z)|+|g(z)|\\leq M$ are also provided. As their applications, Landau's theorems for certain biharmonic mappings are provided, which improve and refine the related results of earlier authors."},"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":"1511.04642","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-11-15T02:13:03Z","cross_cats_sorted":[],"title_canon_sha256":"b93997df20b9e149e1108e04ed23d4b81bc5f82aba2ee59d2505e097de92a123","abstract_canon_sha256":"b0dcd67b4af96ebf67e00400ff38d7a9054eb0297052e2083658c29e788ab25a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:52.146262Z","signature_b64":"IuobAcQieYdC2aCRpXehjH3J4jSeSMMaZP6YRfh4o1tYXY3/PhdEkQE510DG2/pTOwba5zJfsrrwwMm4c1cFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4bdc0ece53f400e7f98d632d7a786d25ccd4cf11ace6dd7dcb3d922b72a6a05f","last_reissued_at":"2026-05-18T01:26:52.145647Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:52.145647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Landau's theorems for certain biharmonic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Ming-Sheng Liu, Yu-Can Zhu, Zhi-Wen Liu","submitted_at":"2015-11-15T02:13:03Z","abstract_excerpt":"Let $f(z)=h(z)+\\overline{g(z)}$ be a harmonic mapping of the unit disk $U$. In this paper, the sharp coefficient estimates for bounded planar harmonic mappings are established, the sharp coefficient estimates for normalized planar harmonic mappings with $|h(z)|+|g(z)|\\leq M$ are also provided. As their applications, Landau's theorems for certain biharmonic mappings are provided, which improve and refine the related results of earlier authors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04642","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":"1511.04642","created_at":"2026-05-18T01:26:52.145734+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.04642v1","created_at":"2026-05-18T01:26:52.145734+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.04642","created_at":"2026-05-18T01:26:52.145734+00:00"},{"alias_kind":"pith_short_12","alias_value":"JPOA5TST6QAO","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"JPOA5TST6QAOP6MN","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"JPOA5TST","created_at":"2026-05-18T12:29:27.538025+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/JPOA5TST6QAOP6MNMMWXU6DNEX","json":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX.json","graph_json":"https://pith.science/api/pith-number/JPOA5TST6QAOP6MNMMWXU6DNEX/graph.json","events_json":"https://pith.science/api/pith-number/JPOA5TST6QAOP6MNMMWXU6DNEX/events.json","paper":"https://pith.science/paper/JPOA5TST"},"agent_actions":{"view_html":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX","download_json":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX.json","view_paper":"https://pith.science/paper/JPOA5TST","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.04642&json=true","fetch_graph":"https://pith.science/api/pith-number/JPOA5TST6QAOP6MNMMWXU6DNEX/graph.json","fetch_events":"https://pith.science/api/pith-number/JPOA5TST6QAOP6MNMMWXU6DNEX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/action/storage_attestation","attest_author":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/action/author_attestation","sign_citation":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/action/citation_signature","submit_replication":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/action/replication_record"}},"created_at":"2026-05-18T01:26:52.145734+00:00","updated_at":"2026-05-18T01:26:52.145734+00:00"}