{"paper":{"title":"A sharp inequality involving hyperbolic and inverse hyperbolic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Roman Drnov\\v{s}ek","submitted_at":"2018-09-04T08:44:31Z","abstract_excerpt":"We prove that the inequality $$\\cosh \\left( \\mathrm{arcosh}(2 \\cosh u) \\cdot \\tanh u \\right) < \\exp \\left( u \\cdot \\tanh u \\right)$$ holds for all $u > 0$. We check with the computation program Mathematica that the ratio between the left-hand and the right-hand side is greater than 0,97 for all $u \\ge 0$, so this is a quite sharp inequality. It is also equivalent to any of the two inequalities: $$ \\cosh \\left( \\sqrt{1 - \\frac{1}{t^2}} \\cdot \\mathrm{arcosh}\\,{2t} \\right) < \\exp \\left( \\sqrt{1 - \\frac{1}{t^2}} \\cdot \\mathrm{arcosh}\\,{t} \\right) $$ for all $t > 1$, and $$ \\cosh \\left( c \\cdot \\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08974","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"}