{"paper":{"title":"On the failure of Bombieri's conjecture for univalent functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Iason Efraimidis","submitted_at":"2016-12-21T17:25:00Z","abstract_excerpt":"A conjecture of Bombieri states that the coefficients of a normalized univalent function $f$ should satisfy $$ \\liminf_{f\\to K} \\frac{n-{\\rm Re\\,}a_n}{m-{\\rm Re\\,}a_m} = \\min_{t\\in{\\mathbb R}} \\, \\frac{n\\sin t -\\sin(nt)}{m\\sin t -\\sin(mt)}, $$ when $f$ approaches the Koebe function $K(z)=\\frac{z}{(1-z)^2}$. Recently, Leung disproved this conjecture for $n=2$ and for all $m\\geq3$ and, also, for $n=3$ and for all odd $m\\geq5$. Complementing his work we disprove it for all $m>n\\geq2$ which are simultaneously odd or even and, also, for the case when $m$ is odd, $n$ is even and $n\\leq \\frac{m+1}{2}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.07242","kind":"arxiv","version":5},"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"}