{"paper":{"title":"Criteria for bounded valence of harmonic mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Juha-Matti Huusko, Mar\\'ia J. Mart\\'in","submitted_at":"2016-11-17T13:01:42Z","abstract_excerpt":"In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative $S(f)$ of a locally univalent analytic function $f$ in the unit disk satisfies that $\\limsup_{|z|\\to 1} |S(f)(z)| (1-|z|^2)^2 < 2$, then there exists a positive integer $N$ such that $f$ takes every value at most $N$ times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function $f$ satisfies that $\\limsup_{|z|\\to 1} |f\"(z)/f'(z)|\\, (1-|z|^2)< 1$.\n  In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when $f$ is merely harmon"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05667","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"}