{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JPOA5TST6QAOP6MNMMWXU6DNEX","short_pith_number":"pith:JPOA5TST","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"},"canonical_sha256":"4bdc0ece53f400e7f98d632d7a786d25ccd4cf11ace6dd7dcb3d922b72a6a05f","source":{"kind":"arxiv","id":"1511.04642","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.04642","created_at":"2026-05-18T01:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1511.04642v1","created_at":"2026-05-18T01:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.04642","created_at":"2026-05-18T01:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"JPOA5TST6QAO","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JPOA5TST6QAOP6MN","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JPOA5TST","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JPOA5TST6QAOP6MNMMWXU6DNEX","target":"record","payload":{"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"},"canonical_sha256":"4bdc0ece53f400e7f98d632d7a786d25ccd4cf11ace6dd7dcb3d922b72a6a05f","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"},"source_kind":"arxiv","source_id":"1511.04642","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":"2026-05-18T01:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LQJIARYoFyhl4SDTzfSkKOvNGZgSnwUlmWQPVONY/IoU9MeKzH7XIKKyokfjDi2S3K6wYPbIuxaHMcWjqHHvDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:00:34.359486Z"},"content_sha256":"b6405b5a8409852e83912b3d80ca8338b9b608350456290636ef5e057e40aaf8","schema_version":"1.0","event_id":"sha256:b6405b5a8409852e83912b3d80ca8338b9b608350456290636ef5e057e40aaf8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JPOA5TST6QAOP6MNMMWXU6DNEX","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WlkQp1Pyhu6KjyYz6/TdCE8/Az5+1ydHkb0sm4rDKYXlKGfGigMgAhpf72U8LH0HiPIFYGLzYcQs+kDD7nksCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T01:00:34.360164Z"},"content_sha256":"853fb55efbf5113a38fd8cb05d5524f17632a8b79ea3365ac16b1f5820899783","schema_version":"1.0","event_id":"sha256:853fb55efbf5113a38fd8cb05d5524f17632a8b79ea3365ac16b1f5820899783"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/bundle.json","state_url":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/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":"2026-06-12T01:00:34Z","links":{"resolver":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX","bundle":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/bundle.json","state":"https://pith.science/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JPOA5TST6QAOP6MNMMWXU6DNEX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JPOA5TST6QAOP6MNMMWXU6DNEX","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":"b0dcd67b4af96ebf67e00400ff38d7a9054eb0297052e2083658c29e788ab25a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-11-15T02:13:03Z","title_canon_sha256":"b93997df20b9e149e1108e04ed23d4b81bc5f82aba2ee59d2505e097de92a123"},"schema_version":"1.0","source":{"id":"1511.04642","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.04642","created_at":"2026-05-18T01:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1511.04642v1","created_at":"2026-05-18T01:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.04642","created_at":"2026-05-18T01:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"JPOA5TST6QAO","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JPOA5TST6QAOP6MN","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JPOA5TST","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:853fb55efbf5113a38fd8cb05d5524f17632a8b79ea3365ac16b1f5820899783","target":"graph","created_at":"2026-05-18T01:26:52Z","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":"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.","authors_text":"Ming-Sheng Liu, Yu-Can Zhu, Zhi-Wen Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-11-15T02:13:03Z","title":"Landau's theorems for certain biharmonic mappings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04642","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:b6405b5a8409852e83912b3d80ca8338b9b608350456290636ef5e057e40aaf8","target":"record","created_at":"2026-05-18T01:26:52Z","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":"b0dcd67b4af96ebf67e00400ff38d7a9054eb0297052e2083658c29e788ab25a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-11-15T02:13:03Z","title_canon_sha256":"b93997df20b9e149e1108e04ed23d4b81bc5f82aba2ee59d2505e097de92a123"},"schema_version":"1.0","source":{"id":"1511.04642","kind":"arxiv","version":1}},"canonical_sha256":"4bdc0ece53f400e7f98d632d7a786d25ccd4cf11ace6dd7dcb3d922b72a6a05f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4bdc0ece53f400e7f98d632d7a786d25ccd4cf11ace6dd7dcb3d922b72a6a05f","first_computed_at":"2026-05-18T01:26:52.145647Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:52.145647Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IuobAcQieYdC2aCRpXehjH3J4jSeSMMaZP6YRfh4o1tYXY3/PhdEkQE510DG2/pTOwba5zJfsrrwwMm4c1cFAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:52.146262Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.04642","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b6405b5a8409852e83912b3d80ca8338b9b608350456290636ef5e057e40aaf8","sha256:853fb55efbf5113a38fd8cb05d5524f17632a8b79ea3365ac16b1f5820899783"],"state_sha256":"9065be8bb0a97237a4371b06e1df9d1d7339786ad30470300631c7ecb973c926"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BPlNfntltghS+upKqJy70xkO/yxerGkfu/z5/YuNtO6Os8y9BpQrIUZMoBHUpuU5XWuMy2jjK7yksMLAaNqfDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T01:00:34.364931Z","bundle_sha256":"a996b0c016d8d1709d54fa76720a2e45ac004477c4e39ad39ffed2f058709699"}}