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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 $$ \\cosh \\left( c \\cdot \\ma"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1809.08974","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-09-04T08:44:31Z","cross_cats_sorted":[],"title_canon_sha256":"ce07ffcc466cff773c0f0e8e49cdd3e3d0e747c9c3ab0473580d3cb6e6179427","abstract_canon_sha256":"312f6c63cbe5e1c2c10ed3c9e9df767f413e8f41ed8e7f4b401a520b491b22cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:03.032107Z","signature_b64":"WEdqDONIyAyJPt2uN6Vhi9g2AFtrNKLwdnMReIHE0+OmLPqSGT46BzfDQiRY6OeybqZ1KnjH9yBNl8Nxi1kUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0939e15d838c16e574eacda8b4eac4faee15c80164ebec01b22873dd4d7f99c7","last_reissued_at":"2026-05-18T00:05:03.031578Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:03.031578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp inequality involving hyperbolic and inverse hyperbolic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Roman Drnov\\v{s}ek","submitted_at":"2018-09-04T08:44:31Z","abstract_excerpt":"We prove that the inequality $$\\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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.08974","created_at":"2026-05-18T00:05:03.031659+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.08974v1","created_at":"2026-05-18T00:05:03.031659+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.08974","created_at":"2026-05-18T00:05:03.031659+00:00"},{"alias_kind":"pith_short_12","alias_value":"BE46CXMDRQLO","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"BE46CXMDRQLOK5HK","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"BE46CXMD","created_at":"2026-05-18T12:32:13.499390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L","json":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L.json","graph_json":"https://pith.science/api/pith-number/BE46CXMDRQLOK5HKZWULJ2WE7L/graph.json","events_json":"https://pith.science/api/pith-number/BE46CXMDRQLOK5HKZWULJ2WE7L/events.json","paper":"https://pith.science/paper/BE46CXMD"},"agent_actions":{"view_html":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L","download_json":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L.json","view_paper":"https://pith.science/paper/BE46CXMD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.08974&json=true","fetch_graph":"https://pith.science/api/pith-number/BE46CXMDRQLOK5HKZWULJ2WE7L/graph.json","fetch_events":"https://pith.science/api/pith-number/BE46CXMDRQLOK5HKZWULJ2WE7L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L/action/storage_attestation","attest_author":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L/action/author_attestation","sign_citation":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L/action/citation_signature","submit_replication":"https://pith.science/pith/BE46CXMDRQLOK5HKZWULJ2WE7L/action/replication_record"}},"created_at":"2026-05-18T00:05:03.031659+00:00","updated_at":"2026-05-18T00:05:03.031659+00:00"}