{"paper":{"title":"Concave univalent functions and Dirichlet finite integral","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":"2015-11-26T07:00:29Z","abstract_excerpt":"The article deals with the class ${\\mathcal F}_{\\alpha }$ consisting of non-vanishing functions $f$ that are analytic and univalent in $\\ID$ such that the complement $\\IC\\backslash f(\\ID) $ is a convex set, $f(1)=\\infty ,$ $f(0)=1$ and the angle at $\\infty $ is less than or equal to $\\alpha \\pi ,$ for some $\\alpha \\in (1,2]$. Related to this class is the class $CO(\\alpha)$ of concave univalent mappings in $\\ID$, but this differs from ${\\mathcal F}_{\\alpha }$ with the standard normalization $f(0)=0=f'(0)=1.$ A number of properties of these classes are discussed which includes an easy proof of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08300","kind":"arxiv","version":2},"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"}