{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:58:54Z","timestamp":1760147934380,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T00:00:00Z","timestamp":1678838400000},"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>Integral inequalities are a powerful tool for estimating errors of quadrature formulas. In this study, some symmetric dual Simpson type integral inequalities for the classes of s-convex, bounded and Lipschitzian functions are proposed. The obtained results are based on a new identity and the use of some standard techniques such as H\u00f6lder as well as power mean inequalities. We give at the end some applications to the estimation of quadrature rules and to particular means.<\/jats:p>","DOI":"10.3390\/sym15030733","type":"journal-article","created":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T06:36:36Z","timestamp":1678862196000},"page":"733","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On s-Convexity of Dual Simpson Type Integral Inequalities"],"prefix":"10.3390","volume":"15","author":[{"given":"Tarek","family":"Chiheb","sequence":"first","affiliation":[{"name":"Laboratory of Analysis and Control of Differential Equations \u201cACED\u201d, Faculty MISM, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hamid","family":"Boulares","sequence":"additional","affiliation":[{"name":"Laboratory of Analysis and Control of Differential Equations \u201cACED\u201d, Faculty MISM, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7684-2616","authenticated-orcid":false,"given":"Moheddine","family":"Imsatfia","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Badreddine","family":"Meftah","sequence":"additional","affiliation":[{"name":"Laboratory of Analysis and Control of Differential Equations \u201cACED\u201d, Faculty MISM, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, Guelma 24000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1080-3686","authenticated-orcid":false,"given":"Abdelkader","family":"Moumen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Ha\u2019il, Ha\u2019il 55425, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,15]]},"reference":[{"key":"ref_1","unstructured":"Pe\u0107arixcx, J.E., Proschan, F., and Tong, Y.L. (1992). Convex Functions, Partial Orderings, and Statistical Applications; Mathematics in Science and Engineering, Academic Press, Inc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"343","DOI":"10.7153\/jmi-02-31","article-title":"General three-point quadrature formulae with applications for \u03b1-L-H\u00f6lder type functions","volume":"2","author":"Bakula","year":"2008","journal-title":"J. Math. Inequal."},{"key":"ref_3","first-page":"511","article-title":"General dual Euler-Simpson formulae","volume":"2","author":"Vukelixcx","year":"2008","journal-title":"J. Math. Inequal."},{"key":"ref_4","first-page":"479","article-title":"On dual Euler-Simpson formulae","volume":"8","author":"Matixcx","year":"2001","journal-title":"Bull. Belg. Math. Soc. Simon Stevin"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"623","DOI":"10.4236\/jamp.2016.44070","article-title":"Schur convexity and the dual Simpson\u2019s formula","volume":"4","author":"Li","year":"2016","journal-title":"J. Appl. Math. Phys."},{"key":"ref_6","first-page":"559","article-title":"Estimations of the error for general Simpson type formulae via pre-Gr\u00fcss inequality","volume":"3","year":"2009","journal-title":"J. Math. Inequal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"53","DOI":"10.5556\/j.tkjm.30.1999.4207","article-title":"On Simpson\u2019s quadrature formula for mappings of bounded variation and applications","volume":"30","author":"Dragomir","year":"1999","journal-title":"Tamkang J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"239","DOI":"10.5556\/j.tkjm.31.2000.398","article-title":"A note on Simpson\u2019s inequality for functions of bounded variation","volume":"31","year":"2000","journal-title":"Tamkang J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1186\/s13662-020-02955-9","article-title":"Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications","volume":"2020","author":"Abdeljawad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_10","first-page":"4","article-title":"New inequalities of Simpson\u2019s type for s-convex functions with applications","volume":"12","author":"Alomari","year":"2009","journal-title":"Res. Rep."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Boulares, H., Meftah, B., Moumen, A., Shafqat, R., Saber, H., Alraqad, T., and Ali, E.E. (2023). Fractional Multiplicative Bullen-Type Inequalities for Multiplicative Differentiable Functions. Symmetry, 15.","DOI":"10.3390\/sym15020451"},{"key":"ref_12","first-page":"409","article-title":"Some extended Simpson-type inequalities and applications","volume":"43","author":"Hsu","year":"2017","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1186\/s13660-018-1924-3","article-title":"Certain new bounds considering the weighted Simpson-like type inequality and applications","volume":"2018","author":"Luo","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1007\/s11253-018-1558-0","article-title":"Simpson-type inequalities for geometrically relative convex functions","volume":"70","author":"Noor","year":"2018","journal-title":"Ukr. Math. J."},{"key":"ref_15","first-page":"13","article-title":"Stetigkeitsaussagen f\u00fcr eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen R\u00e4umen","volume":"23","author":"Breckner","year":"1978","journal-title":"Publ. Inst. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/733\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:55:37Z","timestamp":1760122537000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/733"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,15]]},"references-count":15,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030733"],"URL":"https:\/\/doi.org\/10.3390\/sym15030733","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,3,15]]}}}