{"paper":{"title":"An inequality related to M\\\"{o}bius transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Teerapong Suksumran, Themistocles M. Rassias","submitted_at":"2019-02-13T16:47:24Z","abstract_excerpt":"The open unit ball $\\mathbb{B} = \\{\\mathbf{v}\\in\\mathbb{R}^n\\colon\\|\\mathbf{v}\\|<1\\}$ is endowed with M\\\"{o}bius addition $\\oplus_M$ defined by $$\\mathbf{u}\\oplus_M\\mathbf{v} = \\dfrac{(1 + 2\\langle\\mathbf{u},\\mathbf{v}\\rangle + \\|\\mathbf{v}\\|^2)\\mathbf{u} + (1 - \\|\\mathbf{u}^2)\\mathbf{v}}{1 + \\langle\\mathbf{u},\\mathbf{v}\\rangle + \\|\\mathbf{u}\\|^2\\|\\mathbf{v}\\|^2\\|}$$ for all $\\mathbf{u},\\mathbf{v}\\in \\mathbf{B}$. In this article, we prove the inequality $$ \\dfrac{\\|\\mathbf{u}\\|-\\|\\mathbf{v}\\|}{1+\\|\\mathbf{u}\\|\\|\\mathbf{v}\\|}\\leq \\|\\mathbf{u}\\oplus_M \\mathbf{v}\\| \\leq \\dfrac{\\|\\mathbf{u}\\|+\\|\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.05003","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"}