{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:J4VITNRGENXXVSIWXUKUNOBMF4","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":"73e88f7d7ca55840ef85133589f4877a0e144be1a50976b2d7b3b55222e70e83","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2010-08-28T12:50:46Z","title_canon_sha256":"56c37d6d04639cf0ced75b4ecfc9f17fc70a4e8d6de2733ab944e0f2dad85e30"},"schema_version":"1.0","source":{"id":"1008.4861","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.4861","created_at":"2026-05-18T04:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"1008.4861v1","created_at":"2026-05-18T04:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4861","created_at":"2026-05-18T04:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"J4VITNRGENXX","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"J4VITNRGENXXVSIW","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"J4VITNRG","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:92e8a56244b4148aefde5c8ecd981d4aacee00a0eecbac218954af8532d20bab","target":"graph","created_at":"2026-05-18T04:41:43Z","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 $Co(\\alpha)$ denote the class of concave univalent functions in the unit disk $\\ID$. Each function $f\\in Co(\\alpha)$ maps the unit disk $\\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\\left ( f''(z)/f'(z)\\right)$, $f\\in Co(\\alpha)$. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional $(1-|z|^2)\\left(f''(z)/f'(z)\\right)$, $f\\in Co(\\alpha)$ whenever $f''(0)$ is fixed. We also give a cha","authors_text":"B. Bhowmik, K-J. Wirths, S. Ponnusamy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2010-08-28T12:50:46Z","title":"Characterization and the pre-Schwarzian norm estimate for concave univalent functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4861","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:d133c6600896822592a35b5c2a4ace97d9b1cdd645966b9bc7a1b686feed4f82","target":"record","created_at":"2026-05-18T04:41:43Z","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":"73e88f7d7ca55840ef85133589f4877a0e144be1a50976b2d7b3b55222e70e83","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2010-08-28T12:50:46Z","title_canon_sha256":"56c37d6d04639cf0ced75b4ecfc9f17fc70a4e8d6de2733ab944e0f2dad85e30"},"schema_version":"1.0","source":{"id":"1008.4861","kind":"arxiv","version":1}},"canonical_sha256":"4f2a89b626236f7ac916bd1546b82c2f38aedfa4829bc35465b3979654834744","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4f2a89b626236f7ac916bd1546b82c2f38aedfa4829bc35465b3979654834744","first_computed_at":"2026-05-18T04:41:43.534032Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:43.534032Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RUaq60XaUk5x/xvdZ10wqiDNgjiKzUMzQMrB0JqJt5kSlZKcZ7FhX7lza32SvRkXwrJnNe8Jnrz6dEkTx4GiCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:43.534456Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.4861","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d133c6600896822592a35b5c2a4ace97d9b1cdd645966b9bc7a1b686feed4f82","sha256:92e8a56244b4148aefde5c8ecd981d4aacee00a0eecbac218954af8532d20bab"],"state_sha256":"e5b199bbcecbf3d4ab3b74868a373994ceec86c696282f45c205186ee5a683fd"}