{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:39:32Z","timestamp":1760229572175,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,18]],"date-time":"2022-06-18T00:00:00Z","timestamp":1655510400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Sharp bounds for cosh(x)x,sinh(x)x, and sin(x)x were obtained, as well as one new bound for ex+arctan(x)x. A new situation to note about the obtained boundaries is the symmetry in the upper and lower boundary, where the upper boundary differs by a constant from the lower boundary. New consequences of the inequalities were obtained in terms of the Riemann\u2013Liovuille fractional integral and in terms of the standard integral.<\/jats:p>","DOI":"10.3390\/sym14061260","type":"journal-article","created":{"date-parts":[[2022,6,19]],"date-time":"2022-06-19T21:19:26Z","timestamp":1655673566000},"page":"1260","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4244-4342","authenticated-orcid":false,"given":"Vuk","family":"Stojiljkovi\u0107","sequence":"first","affiliation":[{"name":"Faculty of Science, University of Novi Sad, Trg Dositeja Obradovi\u0107a 3, 21000 Novi Sad, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1845-4441","authenticated-orcid":false,"given":"Slobodan","family":"Radojevi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2818-3174","authenticated-orcid":false,"given":"Ey\u00fcp","family":"\u00c7etin","sequence":"additional","affiliation":[{"name":"Laboratory for Industrial and Applied Mathematics, York University, Toronto, ON M3J 1P3, Canada"},{"name":"New York Business Global, 9591 Baltimore Avenue 703, College Park, MD 20741, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0759-0686","authenticated-orcid":false,"given":"Vesna \u0160e\u0161um","family":"\u010cavi\u0107","sequence":"additional","affiliation":[{"name":"Gra\u0111evinski Fakultet, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8254-6688","authenticated-orcid":false,"given":"Stojan","family":"Radenovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D.S. (1970). Analytic Inequalities, Springer.","DOI":"10.1007\/978-3-642-99970-3"},{"key":"ref_2","unstructured":"Bullen, P.S. (1998). A Dictionary of Inequalities. Pitman Monographs and Surveys in Pure and Applied Mathematics, Addison Wesley Longman Limited."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1090\/S0002-9947-1944-0010188-2","article-title":"Approximation by integral functions in the complex domain","volume":"56","author":"Kober","year":"1944","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_4","first-page":"406","article-title":"On the concavity of sin x\/x","volume":"13","year":"2005","journal-title":"Octogon Math. Mag."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bagul, Y.J., Dhaigude, R.M., Kosti\u0107, M., and Chesneau, C. (2021). Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions. Axioms, 10.","DOI":"10.3390\/axioms10040308"},{"key":"ref_6","first-page":"193","article-title":"Some refinements of well-known inequalities involving trigonometric functions","volume":"36","author":"Chouikla","year":"2021","journal-title":"J. Ramanujan Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2250012","DOI":"10.1142\/S1793557122500127","article-title":"Generalized bounds for sine and cosine functions","volume":"15","author":"Bagul","year":"2022","journal-title":"Asian-Eur. J. Math."},{"key":"ref_8","unstructured":"Dhaigude, M.R., and Yogesh, J.B. (2021). Simple efficient bounds for arcsine and arctangent functions. Punjab Univ. J. Math."},{"key":"ref_9","first-page":"1","article-title":"Refinements and generalizations of certain inequalities involving trigonometric and hyperbolic functions","volume":"1","author":"Neuman","year":"2012","journal-title":"Adv. Inequal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Rodi\u0107, M. (2022). On the Converse Jensen-Type Inequality for Generalized f-Divergences and Zipf\u2013Mandelbrot Law. Mathematics, 10.","DOI":"10.3390\/math10060947"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Rodi\u0107, M. (2022). Some Generalizations of the Jensen-Type Inequalities with Applications. Axioms, 11.","DOI":"10.3390\/axioms11050227"},{"key":"ref_12","unstructured":"Abramowitz, M., and Stegun, I.A. (1992). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover Publications."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hermann, R. (2011). Fractional Calculus An Introduction For Physicists, World Scientific Publishing Co. Pte. Ltd.","DOI":"10.1142\/9789814340250"},{"key":"ref_14","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus Theory and Applications of Differentation and Integration to Arbitrary Order, Academic Press, Inc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Yang, X.J. (2019). General Fractional Derivatives Theory, Methods and Applications, Taylor and Francis Group.","DOI":"10.1201\/9780429284083"},{"key":"ref_16","unstructured":"Anderson, G.D., Vamanamurthy, M.K., and Vuorinen, M. (1997). Conformal Invariants, Inequalities and Quasiconformal Maps, John Wiley and Sons."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1260\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:34:26Z","timestamp":1760139266000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1260"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,18]]},"references-count":16,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061260"],"URL":"https:\/\/doi.org\/10.3390\/sym14061260","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,6,18]]}}}