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We consider the family $Co(p)$ of functions $f:\\ID\\to \\overline{\\IC}$ that satisfy the following conditions: \\bee \\item[(i)] $f$ is meromorphic in $\\ID$ and has a simple pole at the point $p$. \\item[(ii)] $f(0)=f'(0)-1=0$. \\item[(iii)] $f$ maps $\\ID$ conformally onto a set whose complement with respect to $\\overline{\\IC}$ is convex. \\eee We determine the exact domains of variability of some coefficients $a_n(f)$ of the Laurent expansion $$f(z)=\\sum_{n=-1}^{\\infty} a_n(f)(z-p)^n,\\quad |z-p|<1-p, $$ for $f\\in Co(p)$ and certain values of "},"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":"1008.4859","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2010-08-28T12:41:57Z","cross_cats_sorted":[],"title_canon_sha256":"56ce65077de081f723dabf4a194c702cfe51ca50a59eead20138fb4a27e8b08f","abstract_canon_sha256":"29c8cd7360587a42e3dfe79cde6dd3776d468978413b0ac15e73d2f75d7cf118"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:43.545990Z","signature_b64":"K7f9KFgUmVdeobrCtWfTIp14wJ8skEKI5VY7nNWOhYagwdwI1AE4OQ5KymXKDqCphFdRUx1NeMu2EIHxHTsxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c57349bc5e8e3de36eb9847ba95a93d847bc8b261234c6ba1f4539b7026f54b","last_reissued_at":"2026-05-18T04:41:43.545575Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:43.545575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"B. Bhowmik, K-J. Wirths, S. Ponnusamy","submitted_at":"2010-08-28T12:41:57Z","abstract_excerpt":"Let $\\ID$ denote the open unit disc and let $p\\in (0,1)$. We consider the family $Co(p)$ of functions $f:\\ID\\to \\overline{\\IC}$ that satisfy the following conditions: \\bee \\item[(i)] $f$ is meromorphic in $\\ID$ and has a simple pole at the point $p$. \\item[(ii)] $f(0)=f'(0)-1=0$. \\item[(iii)] $f$ maps $\\ID$ conformally onto a set whose complement with respect to $\\overline{\\IC}$ is convex. \\eee We determine the exact domains of variability of some coefficients $a_n(f)$ of the Laurent expansion $$f(z)=\\sum_{n=-1}^{\\infty} a_n(f)(z-p)^n,\\quad |z-p|<1-p, $$ for $f\\in Co(p)$ and certain values of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4859","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":"1008.4859","created_at":"2026-05-18T04:41:43.545636+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.4859v1","created_at":"2026-05-18T04:41:43.545636+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4859","created_at":"2026-05-18T04:41:43.545636+00:00"},{"alias_kind":"pith_short_12","alias_value":"HRLTJG6F5DR5","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_16","alias_value":"HRLTJG6F5DR54NXL","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_8","alias_value":"HRLTJG6F","created_at":"2026-05-18T12:26:07.630475+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/HRLTJG6F5DR54NXLTBD3VFNJHW","json":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW.json","graph_json":"https://pith.science/api/pith-number/HRLTJG6F5DR54NXLTBD3VFNJHW/graph.json","events_json":"https://pith.science/api/pith-number/HRLTJG6F5DR54NXLTBD3VFNJHW/events.json","paper":"https://pith.science/paper/HRLTJG6F"},"agent_actions":{"view_html":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW","download_json":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW.json","view_paper":"https://pith.science/paper/HRLTJG6F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.4859&json=true","fetch_graph":"https://pith.science/api/pith-number/HRLTJG6F5DR54NXLTBD3VFNJHW/graph.json","fetch_events":"https://pith.science/api/pith-number/HRLTJG6F5DR54NXLTBD3VFNJHW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW/action/storage_attestation","attest_author":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW/action/author_attestation","sign_citation":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW/action/citation_signature","submit_replication":"https://pith.science/pith/HRLTJG6F5DR54NXLTBD3VFNJHW/action/replication_record"}},"created_at":"2026-05-18T04:41:43.545636+00:00","updated_at":"2026-05-18T04:41:43.545636+00:00"}