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We prove that such convolutions are locally univalent for $n=1$, which solves an open problem of Dorff et. al (see \\cite[Problem~3.26]{Bshouty2010}). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for $n\\geq 2$."},"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":"1609.05282","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-09-17T04:31:04Z","cross_cats_sorted":[],"title_canon_sha256":"266a02fac20420994f595615bbb6e1ae67d9b2eefdbba12a90549769769ae454","abstract_canon_sha256":"0c59c2987580daff73485768abf5ed3922c9ef6f1b6bef7ac75e29ffd7767b49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:26.852114Z","signature_b64":"mDrLaCkutAWN3xRFYkNgvW+zKCl3Xd+sVPJQhvssSguag8sTP7HUIrL9P5dkyx4xjb6qiki/jOsvqY04sMilAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b1be43b86b251d9f2bb1da2d1d4d08cac41ea8836f181859f2356fde956934e","last_reissued_at":"2026-05-18T01:04:26.851419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:26.851419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Univalency of convolutions of univalent harmonic right half-plane mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Saminathan Ponnusamy, Zhihong Liu","submitted_at":"2016-09-17T04:31:04Z","abstract_excerpt":"We consider the convolution of half-plane harmonic mappings with respective dilatations $(z+a)/(1+az)$ and $e^{i\\theta}z^{n}$, where $-1<a<1$ and $\\theta\\in\\mathbb{R},n\\in\\mathbb{N}$. We prove that such convolutions are locally univalent for $n=1$, which solves an open problem of Dorff et. al (see \\cite[Problem~3.26]{Bshouty2010}). 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