{"paper":{"title":"Extreme points method and univalent harmonic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"S. Ponnusamy, Y. Abu Muhanna","submitted_at":"2014-12-24T12:34:18Z","abstract_excerpt":"We consider the class of all sense-preserving complex-valued harmonic mappings $f=h+\\bar {g}$ defined on the unit disk $\\ID$ with the normalization $h(0)=h'(0)-1=0$ and $g(0)=g'(0)=0$ with the second complex dilatation $\\omega:\\,\\ID\\rightarrow \\ID$, $g'(z)=\\omega (z)h'(z)$. In this paper, the authors determine sufficient conditions on $h$ and $\\omega$ that would imply the univalence of harmonic mappings $f=h+\\bar {g}$ on $\\ID$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.7652","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"}