{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:58:35Z","timestamp":1760144315800,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,9]],"date-time":"2024-04-09T00:00:00Z","timestamp":1712620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Pontificia Universidad Cat\u00f3lica del Ecuador","award":["070-UIO-2022"],"award-info":[{"award-number":["070-UIO-2022"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study uses Raina\u2019s function to obtain a new coordinated pq-integral identity. Using this identity, we construct several new pq-Simpson\u2019s type inequalities for generalized convex functions on coordinates. Setting p1=p2=1 in these inequalities yields well-known quantum Simpson\u2019s type inequalities for coordinated generalized convex functions. Our results have important implications for the creation of post quantum mathematical frameworks.<\/jats:p>","DOI":"10.3390\/sym16040457","type":"journal-article","created":{"date-parts":[[2024,4,10]],"date-time":"2024-04-10T03:07:48Z","timestamp":1712718468000},"page":"457","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["pq-Simpson\u2019s Type Inequalities Involving Generalized Convexity and Raina\u2019s Function"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"first","affiliation":[{"name":"Escuela de Ciencias F\u00edsicas y Matem\u00e1ticas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Cat\u00f3lica del Ecuador, Av. 12 de Octubre 1076, Quito 17-01-2184, Ecuador"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5088-834X","authenticated-orcid":false,"given":"Ghulam Murtaza","family":"Baig","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Management and Technology C-II, Lahore 54700, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1019-9485","authenticated-orcid":false,"given":"Muhammad Uzair","family":"Awan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kamel","family":"Brahim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,9]]},"reference":[{"key":"ref_1","first-page":"533","article-title":"On Simpson\u2019s inequality and applications","volume":"5","author":"Dragomir","year":"2000","journal-title":"J. Inequal. Appl."},{"key":"ref_2","unstructured":"Pe\u010daric, J.E., Proschan, F., and Tong, Y.L. (1992). Convex Functions, Partial Orderings and Statistical Applications, Academic Press."},{"key":"ref_3","first-page":"1","article-title":"New inequalities of Simpson\u2019s type for s-convex functions with applications","volume":"12","author":"Alomari","year":"2009","journal-title":"RGMIA Res. Rep. Coll."},{"key":"ref_4","first-page":"2","article-title":"On new inequalities of Simpson\u2019s type for convex functions","volume":"13","author":"Sarikaya","year":"2010","journal-title":"RGMIA Res. Rep. Coll."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1007\/s13398-020-00841-3","article-title":"On generalizations of some inequalities for convex functions via quantum integrals","volume":"114","author":"Erden","year":"2020","journal-title":"RACSAM"},{"key":"ref_6","unstructured":"\u00d6zdemir, M.E., Akdemir, A.O., Kavurmaci, H., and Avci, M. (2010). On the Simpson\u2019s inequality for coordinated convex functions. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ernst, T.A. (2012). Comprehensive Treatment of q-Calculus, Springer.","DOI":"10.1007\/978-3-0348-0431-8"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M.J., Liko, R., Kashuri, A., and Hern\u00e1ndez Hern\u00e1ndez, J.E. (2019). New Quantum Estimates of Trapezium-Type Inequalities for Generalized \u03d5-Convex Functions. Mathematics, 7.","DOI":"10.3390\/math7111047"},{"key":"ref_9","first-page":"191","article-title":"On generalized Wright\u2019s hypergeometric functions and fractional calculus operators","volume":"21","author":"Raina","year":"2015","journal-title":"East Asian Math. J."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bokulich, A., and Jaeger, G. (2010). Philosophy of Quantum Information Theory and Entaglement, Cambridge University Press.","DOI":"10.1017\/CBO9780511676550"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Zaini, H.G., Treanta, S., Soliman, M.S., and Nanlaopon, K. (2022). Riemann-Liouville Fractional Integral Inequalities for Generalized Pre=Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation. Mathematics, 10.","DOI":"10.3390\/math10020204"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Treanta, S., Soliman, M.S., Zaini, H.G., and Nanlaopon, K. (2022). Some Hadamard-Fejer Type Inequalities for LR-Convex Interval-Valued Functions. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6040178"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2001). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Benatti, F., Fannes, M., Floreanini, R., and Petritis, D. (2010). Quantum Information, Computation and Cryptography: An Introductory Survey of Theory, Technology and Experiments, Springer Science and Business Media.","DOI":"10.1007\/978-3-642-11914-9"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1186\/1687-1847-2013-282","article-title":"Quantum calculus on finite intervals and applications to impulsive difference equations","volume":"2013","author":"Tariboon","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1007\/s10474-020-01025-6","article-title":"On q-Hermite\u2013Hadamard inequalities for general convex functions","volume":"162","author":"Bermudo","year":"2020","journal-title":"Acta Math. Hung."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s00025-018-0783-z","article-title":"On the fundamental Theorem of (p, q)-calculus and some (p, q)-Taylor formulas","volume":"73","author":"Sadjang","year":"2018","journal-title":"Results Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1186\/s13662-020-2512-7","article-title":"On Fractional (p, q)-Calculus","volume":"2020","author":"Soontharanon","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","first-page":"1","article-title":"Some integral inequalities via (p, q)-calculus on finite intervals","volume":"19","year":"2016","journal-title":"RGMIA Res. Rep. Coll."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1186\/s13662-020-03163-1","article-title":"Quantum Hermite\u2013Hadamard-type inequalities for functions with convex absolute values of second qb-derivatives","volume":"2021","author":"Ali","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1515\/math-2021-0015","article-title":"On some new quantum midpoint type inequalities for twice quantum differentiable convex functions","volume":"19","author":"Ali","year":"2021","journal-title":"Open Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.jksus.2016.09.007","article-title":"q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions","volume":"30","author":"Alp","year":"2018","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"199","DOI":"10.22199\/issn.0717-6279-2021-01-0013","article-title":"Some trapezoid and midpoint type inequalities for newly defined quantum integrals","volume":"40","author":"Budak","year":"2021","journal-title":"Proyecciones"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s10957-020-01726-6","article-title":"Some new quantum Hermite\u2013Hadamard-like inequalities for coordinated convex functions","volume":"186","author":"Budak","year":"2020","journal-title":"J. Optim. Theory Appl."},{"key":"ref_25","first-page":"675","article-title":"Some quantum estimates for Hermite\u2013Hadamard inequalities","volume":"251","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Zuo, X., Butt, S.I., Umar, M., Budak, H., and Ali, M.A. (2023). Novel q-differentiable inequalities. Symmetry, 15.","DOI":"10.3390\/sym15081576"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Sitthiwirattham, T., Ali, M.A., and Budak, H. (2023). On some new Maclaurin\u2019s type inequalities for convex functions in q-calculus. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7080572"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1186\/s13662-019-2358-z","article-title":"New parameterized quantum integral inequalities via \u03b7-quasiconvexity","volume":"2019","author":"Nwaeze","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1186\/s13662-020-02559-3","article-title":"Quantum Hermite\u2013Hadamard inequality by means of a Green function","volume":"2020","author":"Khan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4515","DOI":"10.1002\/mma.7048","article-title":"Some new Simpson\u2019s type inequalities for co-ordinated convex functions in quantum calculus","volume":"44","author":"Ali","year":"2021","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1002\/mma.6742","article-title":"Simpson and Newton type inequalities for convex functions via newly defined quantum integrals","volume":"44","author":"Budak","year":"2020","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_32","unstructured":"Rovelli, C. (2004). Quantum Gravity (Cambridge Monographs on Mathematical Physics), Cambridge University Press."},{"key":"ref_33","first-page":"34","article-title":"Fractional calculus applied in solving instability phenomenon in fluid dynamics","volume":"6","author":"Sengar","year":"2015","journal-title":"Int. J. Civ. Eng. Technol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1186\/s13662-020-03195-7","article-title":"Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables","volume":"2021","author":"Ali","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1515\/math-2021-0020","article-title":"Quantum Ostrowski type inequalities for twice quantum differentiable functions in quantum calculus","volume":"19","author":"Ali","year":"2021","journal-title":"Open Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s13398-017-0402-y","article-title":"(p, q)-Hermite\u2013Hadamard inequalities and (p, q)-estimates for midpoint inequalities via convex quasi-convex functions","volume":"112","author":"Kunt","year":"2018","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00edsicas Nat. Ser. A Matem\u00c1ticas"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4011","DOI":"10.3934\/math.2020258","article-title":"Post-quantum trapezoid type inequalities","volume":"5","author":"Latif","year":"2020","journal-title":"AIMS Math."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., Murtaza, G., Baig, G.M., and Awan, M.U. (2023). Raina\u2019s Function-Based Formulations of Right-Sided Simpson\u2019s and Newton\u2019s Inequalities for Generalized Coordinated Convex Functions. Symmetry, 15.","DOI":"10.3390\/sym15071441"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1214\/aos\/1032894463","article-title":"Some inequalities for symmetric convex sets with applications","volume":"24","author":"Anderson","year":"1996","journal-title":"Ann. Stat."},{"key":"ref_40","first-page":"345","article-title":"Centrally symmetric convex sets","volume":"14","author":"Boltyanski","year":"2007","journal-title":"J. Convex Anal."},{"key":"ref_41","first-page":"193","article-title":"On a q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1186\/s13662-020-03094-x","article-title":"New post-quantum analogues of Ostrowski-type inequalities using new definitions of left-right (p,q)-derivatives and definite integrals","volume":"2020","author":"Chu","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., Ali, M.A., Budak, H., Kalsoom, H., and Agarwal, P. (2021). Some New Hermite\u2013Hadamard and Related Inequalities for Convex Functions via (p,q)-Integral. Entropy, 23.","DOI":"10.3390\/e23070828"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/457\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:25:22Z","timestamp":1760106322000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/457"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,9]]},"references-count":43,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["sym16040457"],"URL":"https:\/\/doi.org\/10.3390\/sym16040457","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,4,9]]}}}