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As a consequence of the results, we prove a conjecture due to Bhayo and Vuorinen [J. Approx. Theory, 164(2012)]."},"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":"1212.4681","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-12-05T05:23:51Z","cross_cats_sorted":[],"title_canon_sha256":"0e8406859eafe7b674f998de1954273325fd565fb4982e2060dd2d71736e74bf","abstract_canon_sha256":"7712ee6eda685183bef2346c2fad797827fce848c98087e95e4484df3ddc76ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:11.759550Z","signature_b64":"HuLwxPb3pYeDGypA4iWUHNPbWu0xKAmrxTE+aZpA8cRQ5aIssW62L9ZOqCAwl4YkLYOIN4/NvfIyzIcK3M7kBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3059511cfe12e425ecfdd8abffe9e802b4e62ee59efccee8c1696b24c8853121","last_reissued_at":"2026-05-18T03:38:11.759161Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:11.759161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Inequalities for generalized trigonometric and hyperbolic sine functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Miao-Kun Wang, Yue-ping Jiang, Yu-Ming Chu","submitted_at":"2012-12-05T05:23:51Z","abstract_excerpt":"We prove that the inequalities $\\sin_{p,q}(\\sqrt{rs})\\geq \\sqrt{\\sin_{p,q}(r)\\sin_{p,q}(s)}$ and $\\sinh_{p,q}(\\sqrt{r^*s^*}) \\leq \\sqrt{\\sinh_{p,q}(r^*)\\sinh_{p,q}(s^*)}$ hold for all $p,q\\in(1,\\infty)$, $r,s\\in(0,\\int_{0}^{1}(1-t^q)^{-1/p}dt)$ and $r^*,s^*\\in(0,\\int_{0}^{\\infty}(1+t^q)^{-1/p}dt)$, where $\\sin_{p,q}$ and $\\sinh_{p,q}$ are the generalized trigonometric and hyperbolic sine functions, respectively. 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