{"paper":{"title":"Sharp Bounds for the Arc Lemniscate Sine Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Horst Alzer, Man Kam Kwong","submitted_at":"2019-03-10T01:17:20Z","abstract_excerpt":"The arc lemniscate sine function is given by $$ \\mbox{arcsl}(x)=\\int_0^x \\frac{1}{\\sqrt{1-t^4}}dt. $$ In 2017, Mahmoud and Agarwal presented bounds for $\\mbox{arcsl}$ in terms of the Lerch zeta function $$ \\Phi(z,s,a)=\\sum_{k=0}^\\infty \\frac {z^k}{(k+a)^s}. $$ They proved $$ \\frac{1}{8} \\, x \\, \\Phi(x^4, 3/2, 1/4) < \\mbox{arcsl}(x)< \\frac{1}{4} \\, x \\, \\Phi(x^4,3/2,1/4)\\qquad{(0<x<1)}. $$ We %use the monotone form of l'Hopital's rule to show that the factor $1/4$ can be replaced by $\\mbox{arcsl}(1)/\\Phi(1,3/2,1/4)=0.12836...$. This constant is best possible."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.03897","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"}