{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T05:05:30Z","timestamp":1777352730400,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T00:00:00Z","timestamp":1637193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1\u2212\u03b1x2)e\u03b2x2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0,\u00a0\u03c0), while our main result for the cosine function is a double inequality holding on the interval (0,\u00a0\u03c0\/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l\u2019Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.<\/jats:p>","DOI":"10.3390\/axioms10040308","type":"journal-article","created":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T08:53:56Z","timestamp":1637225636000},"page":"308","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8331-3920","authenticated-orcid":false,"given":"Yogesh J.","family":"Bagul","sequence":"first","affiliation":[{"name":"Department of Mathematics, K. K. M. College, Manwath 431505, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ramkrishna M.","family":"Dhaigude","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati 444604, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marko","family":"Kosti\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovi\u0107a 6, 21125 Novi Sad, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Chesneau","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Caen-Normandie, 14032 Caen, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,18]]},"reference":[{"key":"ref_1","unstructured":"Abramowitz, M., and Stegun, I.A. (1972). 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