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Let $a_n,b_n, A_n, B_n$ denote the Maclaurin coefficients of $h,g,H,G$, and $${\\mathcal F}=\\{f=h+\\bar{g}\\in {\\mathcal H}:\\,|a_n|\\leq A_n and |b_n|\\leq B_n for n\\geq 1}. $$ We show that the radius of univalence of the family ${\\mathcal F}$ is $0.112903...$. We also show that this number is also the radius of the starlikeness of ${\\mathcal F}$. Analogous results are proved for a subclass of the class of har"},"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":"1107.0610","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-07-04T12:38:02Z","cross_cats_sorted":[],"title_canon_sha256":"00002e684d0c41388d994d61b526668130374d447a5ff173bbdeb747e72c61bb","abstract_canon_sha256":"809cf288aa0d789a0e546350c30dfd0d068d98fc0b87c703b2501c76181a166d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:53.696243Z","signature_b64":"mPGGcaJdP+EP/DKJcZL670+BPy9ZQNqWHgtclDHhK8R2ybIjUvNQQryuoTiQagBTrowaeYoy4rDbiSb4U+3CBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a215b70cc723c2b53c0e1b7c2fa5ee0a22d500b1132d3908b27812eb6b25b26","last_reissued_at":"2026-05-18T04:18:53.695848Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:53.695848Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Radius of Close-to-convexity of Harmonic Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"David Kalaj, Matti Vuorinen, Saminathan Ponnusamy","submitted_at":"2011-07-04T12:38:02Z","abstract_excerpt":"Let ${\\mathcal H}$ denote the class of all normalized complex-valued harmonic functions $f=h+\\bar{g}$ in the unit disk ${\\mathbb D}$, and let $K=H+\\bar{G}$ denote the harmonic Koebe function. Let $a_n,b_n, A_n, B_n$ denote the Maclaurin coefficients of $h,g,H,G$, and $${\\mathcal F}=\\{f=h+\\bar{g}\\in {\\mathcal H}:\\,|a_n|\\leq A_n and |b_n|\\leq B_n for n\\geq 1}. $$ We show that the radius of univalence of the family ${\\mathcal F}$ is $0.112903...$. We also show that this number is also the radius of the starlikeness of ${\\mathcal F}$. 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