{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T05:02:02Z","timestamp":1778043722747,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T00:00:00Z","timestamp":1663286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62002079"],"award-info":[{"award-number":["62002079"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, developments and extensions of quadrature inequalities in quantum calculus have been extensively studied. As a result, several quantum extensions of Simpson\u2019s and Newton\u2019s estimates are examined in order to explore different directions in quantum studies. The main motivation of this article is the development of variants of Simpson\u2013Newton-like inequalities by employing Mercer\u2019s convexity in the context of quantum calculus. The results also give new quantum bounds for Simpson\u2013Newton-like inequalities through H\u00f6lder\u2019s inequality and the power mean inequality by employing the Mercer scheme. The validity of our main results is justified by providing examples with graphical representations thereof. The obtained results recapture the discoveries of numerous authors in quantum and classical calculus. Hence, the results of these inequalities lead us to the development of new perspectives and extensions of prior results.<\/jats:p>","DOI":"10.3390\/sym14091935","type":"journal-article","created":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T04:28:55Z","timestamp":1663648135000},"page":"1935","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["New Quantum Mercer Estimates of Simpson\u2013Newton-like Inequalities via Convexity"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7192-8269","authenticated-orcid":false,"given":"Saad","family":"Ihsan Butt","sequence":"first","affiliation":[{"name":"Department of Mathematics, Lahore Campus, COMSATS University Islamabad, Islamabad 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8843-955X","authenticated-orcid":false,"given":"H\u00fcseyin","family":"Budak","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, D\u00fczce University, 81620 D\u00fczce, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D.S., Pe\u010dari\u0107, J.E., and Fink, A.M. 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