{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:25:19Z","timestamp":1760235919494,"version":"build-2065373602"},"reference-count":60,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,3]],"date-time":"2021-10-03T00:00:00Z","timestamp":1633219200000},"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-21"],"award-info":[{"award-number":["KMUTNB-63-KNOW-21"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we establish some new Hermite\u2013Hadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for p,q-differentiable preinvex functions in the context of p,q-calculus. We also show that the results established in this paper are generalizations of comparable results in the literature of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.<\/jats:p>","DOI":"10.3390\/sym13101864","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1864","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On Some New Inequalities of Hermite\u2013Hadamard Midpoint and Trapezoid Type for Preinvex Functions in p,q-Calculus"],"prefix":"10.3390","volume":"13","author":[{"given":"Ifra Bashir","family":"Sial","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China"}]},{"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":"Ghulam","family":"Murtaza","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology, Lahore 54700, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University,  Jeddah 21589, Saudi Arabia"}]},{"given":"Jarunee","family":"Soontharanon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1402","authenticated-orcid":false,"given":"Thanin","family":"Sitthiwirattham","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,3]]},"reference":[{"key":"ref_1","first-page":"5","article-title":"Fractional non conformable Hermite-Hadamard inequalities for generalized \u03d5-convex functions","volume":"64","author":"Ali","year":"2020","journal-title":"Fasc. 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