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Assume that $F$ is a holomorphic function on $\\Delta^+$ with continuous extension up to $\\gamma$ such that $F$ maps $\\gamma$ into $\\{|\\mbox{Im} z|\\leq C|\\mbox{Re} z|\\},$ for some positive $C.$ If $F$ vanishes to infinite order at $0$ then $F$ vanishes identically. We show that given the conditions of the conjecture, either $F\\equiv 0$ or there is a sequence in $\\Delta^+$, converging to $0,$ alo"},"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":"1407.1763","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-07-07T16:32:07Z","cross_cats_sorted":[],"title_canon_sha256":"ab7112d165fd8582190157818863166983fb76ba7d43379fcfb18e248d8ffc7f","abstract_canon_sha256":"5bfda14da672588fa87a33116a7f3d978e1ba4e68f3a41850d23387147728fa5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:27.126827Z","signature_b64":"E9vIXmI87O56OepwZpt9vtUpspnYIh1GpQWbYF/78Dz08g36uSbOsLZfE0l8qQ/caUosOMINypUCqeJMg6vEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b897504abbd663f17bfd519ae5b46c61b3653da8c0528ef49caa155e00d5120e","last_reissued_at":"2026-05-18T01:35:27.126189Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:27.126189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on a conjecture concerning boundary uniqueness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Abtin Daghighi, Steven G. Krantz","submitted_at":"2014-07-07T16:32:07Z","abstract_excerpt":"We consider the following conjecture (from Huang, et al): Let $\\Delta^+$ denote the upper half disc in $\\mathbb{C}$ and let $\\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\\mathbb{C}$). Assume that $F$ is a holomorphic function on $\\Delta^+$ with continuous extension up to $\\gamma$ such that $F$ maps $\\gamma$ into $\\{|\\mbox{Im} z|\\leq C|\\mbox{Re} z|\\},$ for some positive $C.$ If $F$ vanishes to infinite order at $0$ then $F$ vanishes identically. 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