{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T07:14:14Z","timestamp":1774163654352,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T00:00:00Z","timestamp":1683504000000},"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>The objective of this study is to identify novel quantum midpoint-type inequalities for twice\u00a0q-differentiable functions by utilizing Mercer\u2019s approach. We introduce a new auxiliary variant of the quantum Mercer midpoint-type identity related to twice\u00a0q-differentiable functions. By applying the theory of convex functions to this identity, we introduce new bounds using well-known inequalities, such as H\"older\u2019s inequality and power-mean inequality. We provide explicit examples along with graphical demonstrations. The findings of this study explain previous studies on midpoint-type inequalities. Analytic inequalities of this type, as well as related strategies, have applications in various fields where symmetry plays an important role.<\/jats:p>","DOI":"10.3390\/sym15051038","type":"journal-article","created":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T03:34:53Z","timestamp":1683516893000},"page":"1038","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["New Study on the Quantum Midpoint-Type Inequalities for Twice q-Differentiable Functions via the Jensen\u2013Mercer Inequality"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7192-8269","authenticated-orcid":false,"given":"Saad Ihsan","family":"Butt","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9911-1111","authenticated-orcid":false,"given":"Muhammad","family":"Umar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 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, D\u00fczce 81620, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,8]]},"reference":[{"key":"ref_1","unstructured":"Mitrinovi\u0107, D.S., Pexcxarixcx, J.E., and Fink, A.M. 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