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For $\\phi \\in \\mathcal{A}$, let $\\mathcal{W}_{H}^{-}(\\phi):=\\{f=h+\\bar{g} \\in \\mathcal{H}:h-g=\\phi\\}$ and $\\mathcal{W}_{H}^{+}(\\phi):=\\{f=h+\\bar{g} \\in \\mathcal{H}:h+g=\\phi\\}$ be subfamilies of $\\mathcal{H}$. In this paper, we shall determine the conditions under which the harmonic convolution $f_1*f_2$ is univalent and convex in one direction if"},"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":"1302.5791","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-02-23T12:21:19Z","cross_cats_sorted":[],"title_canon_sha256":"b5e20f76b3ec54fb9cde8053f5283d6921e122b6633dbcb2a44ac558e065100f","abstract_canon_sha256":"eadd3b073d1ed415c108b61f975c65640ceb76d54e2a2a0aa6a8e08e4bf2eef2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:43.732231Z","signature_b64":"3UOgbSWHLJ/9+UnDCFXsmJkQVmdN6GfS7PPFt8IJeWZe1bHncWAwQFyy5RYiYnoBQGDsbBUl/9k4SsYrQIMSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c64027a8f0b30a986e1939555135bd48c55b7bdc89492c540d7f730e8fa011e","last_reissued_at":"2026-05-18T03:32:43.731542Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:43.731542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Univalence and convexity in one direction of the convolution of harmonic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Sumit Nagpal, V. Ravichandran","submitted_at":"2013-02-23T12:21:19Z","abstract_excerpt":"Let $\\mathcal{H}$ denote the class of all complex-valued harmonic functions $f$ in the open unit disk normalized by $f(0)=0=f_{z}(0)-1=f_{\\bar{z}}(0)$, and let $\\mathcal{A}$ be the subclass of $\\mathcal{H}$ consisting of normalized analytic functions. For $\\phi \\in \\mathcal{A}$, let $\\mathcal{W}_{H}^{-}(\\phi):=\\{f=h+\\bar{g} \\in \\mathcal{H}:h-g=\\phi\\}$ and $\\mathcal{W}_{H}^{+}(\\phi):=\\{f=h+\\bar{g} \\in \\mathcal{H}:h+g=\\phi\\}$ be subfamilies of $\\mathcal{H}$. In this paper, we shall determine the conditions under which the harmonic convolution $f_1*f_2$ is univalent and convex in one direction if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5791","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":"1302.5791","created_at":"2026-05-18T03:32:43.731650+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.5791v1","created_at":"2026-05-18T03:32:43.731650+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5791","created_at":"2026-05-18T03:32:43.731650+00:00"},{"alias_kind":"pith_short_12","alias_value":"DRSAE6UPBMYK","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"DRSAE6UPBMYKTBXB","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"DRSAE6UP","created_at":"2026-05-18T12:27:43.054852+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/DRSAE6UPBMYKTBXBSOKVKE232S","json":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S.json","graph_json":"https://pith.science/api/pith-number/DRSAE6UPBMYKTBXBSOKVKE232S/graph.json","events_json":"https://pith.science/api/pith-number/DRSAE6UPBMYKTBXBSOKVKE232S/events.json","paper":"https://pith.science/paper/DRSAE6UP"},"agent_actions":{"view_html":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S","download_json":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S.json","view_paper":"https://pith.science/paper/DRSAE6UP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.5791&json=true","fetch_graph":"https://pith.science/api/pith-number/DRSAE6UPBMYKTBXBSOKVKE232S/graph.json","fetch_events":"https://pith.science/api/pith-number/DRSAE6UPBMYKTBXBSOKVKE232S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S/action/storage_attestation","attest_author":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S/action/author_attestation","sign_citation":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S/action/citation_signature","submit_replication":"https://pith.science/pith/DRSAE6UPBMYKTBXBSOKVKE232S/action/replication_record"}},"created_at":"2026-05-18T03:32:43.731650+00:00","updated_at":"2026-05-18T03:32:43.731650+00:00"}