{"paper":{"title":"Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"B. Bhowmik, K-J. Wirths, S. Ponnusamy","submitted_at":"2010-08-28T12:41:57Z","abstract_excerpt":"Let $\\ID$ denote the open unit disc and let $p\\in (0,1)$. We consider the family $Co(p)$ of functions $f:\\ID\\to \\overline{\\IC}$ that satisfy the following conditions: \\bee \\item[(i)] $f$ is meromorphic in $\\ID$ and has a simple pole at the point $p$. \\item[(ii)] $f(0)=f'(0)-1=0$. \\item[(iii)] $f$ maps $\\ID$ conformally onto a set whose complement with respect to $\\overline{\\IC}$ is convex. \\eee We determine the exact domains of variability of some coefficients $a_n(f)$ of the Laurent expansion $$f(z)=\\sum_{n=-1}^{\\infty} a_n(f)(z-p)^n,\\quad |z-p|<1-p, $$ for $f\\in Co(p)$ and certain values of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4859","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"}