{"paper":{"title":"Ahlfors-Beurling conformal invariant and relative capacity of compact sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CV","authors_text":"Matti Vuorinen, Vladimir N. Dubinin","submitted_at":"2011-12-19T05:44:28Z","abstract_excerpt":"For a given domain $D$ in the extended complex plane $\\bar{\\mathbb C}$ with an accessible boundary point $z_0 \\in \\partial D$ and for a subset $E \\subset {D},$ relatively closed w.r.t. $D,$ we define the relative capacity $\\rc E$ as a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal invariant $r(D\\setminus E,z)/r(D, z)$ when $z$ approaches the point $z_0.$ Here $r(G,z)$ denotes the inner radius at $z$ of the connected component of the set $G$ containing the point $z.$ The asymptotic behavior of this quotient is established. Further, it is shown that in the case when th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4245","kind":"arxiv","version":4},"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"}