{"paper":{"title":"Heinz inequality for the unit ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Kalaj","submitted_at":"2015-04-07T17:53:58Z","abstract_excerpt":"We first prove the following generalization of Schwarz lemma for harmonic mappings. Let $u$ be a harmonic mapping of the unit ball onto itself. Then we prove the inequality $\\|u(x)-(1-\\|x\\|^2)/(1+\\|x\\|^2)^{n/2} u(0)\\|\\le U(|x| N)$. By using the Schwarz lemma for harmonic mappings we derive Heinz inequality on the boundary of the unit ball by providing a sharp constant $C_n$ in the inequality: $\\|\\partial_r u(r\\eta)\\|_{r=1}\\ge C_n$, $\\|\\eta\\|=1$, for every harmonic mapping of the unit ball into itself satisfying the condition $u(0)=0$, $\\|u(\\eta)\\|=1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01686","kind":"arxiv","version":2},"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"}