{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:34:26Z","timestamp":1771673666399,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T00:00:00Z","timestamp":1634169600000},"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>Numerical approximations of definite integrals and related error estimations can be made using Simpson\u2019s rules (inequalities). There are two well-known rules: Simpson\u2019s 13 rule or Simpson\u2019s quadrature formula and Simpson\u2019s 38 rule or Simpson\u2019s second formula. The aim of the present paper is to extend several inequalities that hold for Simpson\u2019s 13 rule to Simpson\u2019s 38 rule. More precisely, we prove a weighted version of Simpson\u2019s second type inequality and some Simpson\u2019s second type inequalities for Lipschitzian, bounded variations, convex functions and the functions that belong to Lq. Some applications of the second type Simpson\u2019s inequalities relate to approximations of special means and Simpson\u2019s 38 formula, and moments of random variables are made.<\/jats:p>","DOI":"10.3390\/sym13101933","type":"journal-article","created":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T23:02:16Z","timestamp":1634252536000},"page":"1933","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["On Weighted Simpson\u2019s \r\n  38 Rule"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2461-6630","authenticated-orcid":false,"given":"Mohsen","family":"Rostamian Delavar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord 94531, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0115-3079","authenticated-orcid":false,"given":"Artion","family":"Kashuri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Technical Science, University \u201cIsmail Qemali\u201d, 9400 Vlora, Albania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De La Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes, Campus of Leioa (Bizkaia)-Aptdo, 644-Bilbao, University of Basque Country, 48080 Bilbao, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,14]]},"reference":[{"key":"ref_1","first-page":"533","article-title":"On Simpson\u2019s inequality and applications","volume":"5","author":"Dragomir","year":"2000","journal-title":"J. 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