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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."},"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":"1610.05495","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-10-18T09:20:53Z","cross_cats_sorted":[],"title_canon_sha256":"a60a134a621e13bf0debc14efddc69798e4e5a944538c28b7e4af4a5d608539a","abstract_canon_sha256":"51fe811d073d123f09227d732b548da5b716c3360f2196fa70950bdd7d9eabe3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:59.186879Z","signature_b64":"Wfpub54gNfW3DQ61dUWMtNdwrORog0nwOUILrEGO42WBcGi68tKY9P4ZV1bQyRyZ19mz3rz3/AsEk4VqQzqOBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d8deeb741f0680baae65e9b63bf23f5cf48aaa404af6eb864eb98c0d8268da0","last_reissued_at":"2026-05-18T01:01:59.186253Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:59.186253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $x\\in(0,\\pi)$. 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