{"paper":{"title":"On a Sine Polynomial of Turan","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":"2016-10-18T09:20:53Z","abstract_excerpt":"In 1935, P. Tur\\'an proved that $$ S_{n,a}(x)= \\sum_{j=1}^n{n+a-j\\choose n-j} \\sin(jx)>0 \\quad{(n,a\\in\\mathbf{N}; 0<x<\\pi).} $$ We present various related inequalities. Among others, we show that the refinements $$ S_{2n-1,a}(x)\\geq \\sin(x) \\quad\\mbox{and} \\quad{S_{2n,a}(x)\\geq 2\\sin(x)(1+\\cos(x))} $$ are valid for all integers $n\\geq 1$ and real numbers $a\\geq 1$ and $x\\in(0,\\pi)$. Moreover, we apply our theorems on sine sums to obtain inequalities for the Chebyshev polynomials of the second kind."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05495","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"}