{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T04:09:45Z","timestamp":1771646985507,"version":"3.50.1"},"reference-count":57,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,10]],"date-time":"2020-03-10T00:00:00Z","timestamp":1583798400000},"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>By using the contemporary theory of inequalities, this study is devoted to proposing a number of refinements inequalities for the Hermite-Hadamard\u2019s type inequality and conclude explicit bounds for two new definitions of     (  p 1   p 2  ,  q 1   q 2  )    -differentiable function and     (  p 1   p 2  ,  q 1   q 2  )    -integral for two variables mappings over finite rectangles by using pre-invex set. We have derived a new auxiliary result for     (  p 1   p 2  ,  q 1   q 2  )    -integral. Meanwhile, by using the symmetry of an auxiliary result, it is shown that novel variants of the the Hermite-Hadamard type for     (  p 1   p 2  ,  q 1   q 2  )    -differentiable utilizing new definitions of generalized higher-order strongly pre-invex and quasi-pre-invex mappings. It is to be acknowledged that this research study would develop new possibilities in pre-invex theory, quantum mechanics and special relativity frameworks of varying nature for thorough investigation.<\/jats:p>","DOI":"10.3390\/sym12030443","type":"journal-article","created":{"date-parts":[[2020,3,13]],"date-time":"2020-03-13T08:58:59Z","timestamp":1584089939000},"page":"443","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Post Quantum Integral Inequalities of Hermite-Hadamard-Type Associated with Co-Ordinated Higher-Order Generalized Strongly Pre-Invex and Quasi-Pre-Invex Mappings"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-3349","authenticated-orcid":false,"given":"Humaira","family":"Kalsoom","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7137-1720","authenticated-orcid":false,"given":"Saima","family":"Rashid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}]},{"given":"Muhammad","family":"Idrees","sequence":"additional","affiliation":[{"name":"Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China"}]},{"given":"Farhat","family":"Safdar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, SBK Women\u2019s University, Quetta 87300, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6434-7650","authenticated-orcid":false,"given":"Saima","family":"Akram","sequence":"additional","affiliation":[{"name":"Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 66000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 06530, Turkey"},{"name":"Institute of Space Sciences, 077125 Magurele-Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0944-2134","authenticated-orcid":false,"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,10]]},"reference":[{"key":"ref_1","first-page":"193","article-title":"On a q-definite integrals","volume":"4","author":"Jackson","year":"1910","journal-title":"Quart. 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