{"paper":{"title":"Hyperbolic geodesics, Krzyz's conjecture and beyond","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Samuel L. Krushkal","submitted_at":"2016-03-08T20:48:41Z","abstract_excerpt":"In 1968, Krzyz conjectured that for non-vanishing holomorphic functions $f(z) = c_0 + c_1 z + \\dots$ in the unit disk with $|f(z)| \\leq 1$, we have the sharp bound $|c_n| \\leq 2/e$ for all $n \\geq 1$, with equality only for the function $f(z) = exp [(z^n - 1)/(z^n + 1)]$ and its rotations. This conjecture was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n \\leq 5$.\n  We provide here two different proofs of this conjecture and its generalizations based on completely different ideas."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02668","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"}