{"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. As a consequence of the results, we prove a conjecture due to Bhayo and Vuorinen [J. Approx. Theory, 164(2012)]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4681","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"}