{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:47:53Z","timestamp":1760150873757,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,23]],"date-time":"2022-01-23T00:00:00Z","timestamp":1642896000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007345","name":"King Mongkut's University of Technology North Bangkok","doi-asserted-by":"publisher","award":["KMUTNB-63-KNOW-22"],"award-info":[{"award-number":["KMUTNB-63-KNOW-22"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, first we establish a p,q-integral identity involving the second p,q-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-p,q-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature.<\/jats:p>","DOI":"10.3390\/axioms11020046","type":"journal-article","created":{"date-parts":[[2022,1,23]],"date-time":"2022-01-23T20:32:52Z","timestamp":1642969972000},"page":"46","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Post-Quantum Midpoint-Type Inequalities Associated with Twice-Differentiable Functions"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1402","authenticated-orcid":false,"given":"Thanin","family":"Sitthiwirattham","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"}]},{"given":"Ghulam","family":"Murtaza","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54700, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5341-4926","authenticated-orcid":false,"given":"Muhammad Aamir","family":"Ali","sequence":"additional","affiliation":[{"name":"Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China"}]},{"given":"Chanon","family":"Promsakon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"given":"Ifra Bashir","family":"Sial","sequence":"additional","affiliation":[{"name":"School of Sciences, Jiangsu University, Zhenjiang 212000, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7556-8942","authenticated-orcid":false,"given":"Praveen","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Anand International College of Engineering, Jaipur 302020, India"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,23]]},"reference":[{"key":"ref_1","unstructured":"Pe\u010dari\u0107, J.E., Proschan, F., and Tong, Y.L. 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