{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T16:17:43Z","timestamp":1783959463460,"version":"3.55.0"},"reference-count":35,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,8]],"date-time":"2020-09-08T00:00:00Z","timestamp":1599523200000},"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>Some recent results have been found treating the famous Simpson\u2019s rule in connection with the convexity property of functions and those called generalized convex. The purpose of this article is to address Newton-type integral inequalities by associating with them certain criteria of quantum calculus and the convexity of the functions of various variables. In this article, by using the concept of recently defined q1q2 -derivatives and integrals, some of Newton\u2019s type inequalities for co-ordinated convex functions are revealed. We also employ the limits of q1,q2\u21921\u2212 in new results, and attain some new inequalities of Newton\u2019s type for co-ordinated convex functions through ordinary integral. Finally, we provide a thorough application of the newly obtained key outcomes, these new consequences can be useful in the integral approximation study for symmetrical functions, or with some kind of symmetry.<\/jats:p>","DOI":"10.3390\/sym12091476","type":"journal-article","created":{"date-parts":[[2020,9,10]],"date-time":"2020-09-10T22:58:01Z","timestamp":1599778681000},"page":"1476","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":64,"title":["Some New Newton\u2019s Type Integral Inequalities for Co-Ordinated Convex Functions in Quantum Calculus"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"first","affiliation":[{"name":"Pontificia Universidad Cat\u00f3lica del Ecuador, Facultad de Ciencias Exactas y Naturales, Escuela de Ciencias Matem\u00e1ticas y F\u00edsicas, Av. 12 de octubre 1076, Apartado 17-01-2184, Ecuador"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Aamir Ali","sequence":"additional","affiliation":[{"name":"Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0115-3079","authenticated-orcid":false,"given":"Artion","family":"Kashuri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, 9400 Vlora, Albania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ifra","family":"Bashir Sial","sequence":"additional","affiliation":[{"name":"School of Control Science and Engineering, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiyue","family":"Zhang","sequence":"additional","affiliation":[{"name":"Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,8]]},"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","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_3","first-page":"1","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_4","unstructured":"\u00d6zdemir, M.E., Akdemir, A.O., Kavurmaci, H., and Avci, M. (2010). On the Simpson\u2019s inequality for co-ordinated convex functions. arXiv."},{"key":"ref_5","first-page":"7","article-title":"Some Newton\u2019s type inequalities for harmonic convex functions","volume":"9","author":"Noor","year":"2016","journal-title":"J. Adv. Math. Stud."},{"key":"ref_6","first-page":"239","article-title":"Newton inequalities for p-harmonic convex functions","volume":"40","author":"Noor","year":"2018","journal-title":"Honam Math. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0218348X20500371","article-title":"Newton\u2019s type integral inequalities via local fractional integrals","volume":"28","author":"Iftikhar","year":"2020","journal-title":"Fractals"},{"key":"ref_8","unstructured":"Ernst, T. (2000). The History of q-calculus and a New Method, Department of Mathematics, Uppsala University."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ernst, T. (2012). A Comprehensive Treatment of q-Calculus, Springer.","DOI":"10.1007\/978-3-0348-0431-8"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/S0898-1221(04)90025-9","article-title":"Integral inequalities in q-calculus","volume":"47","author":"Gauchman","year":"2004","journal-title":"Comput. Math. Appl."},{"key":"ref_11","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_12","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2002). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_13","first-page":"1","article-title":"Quantum calculus on finite intervals and applications to impulsive difference equations","volume":"282","author":"Tariboon","year":"2013","journal-title":"Adv. Differ. Equ."},{"key":"ref_14","first-page":"1","article-title":"Simpson\u2019s Integral Inequalities for Twice Differentiable Convex Functions","volume":"2020","author":"Abdeljawad","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.jksus.2016.09.007","article-title":"q\u2013Hermite 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_16","first-page":"341","article-title":"Hermite\u2013Hadamard\u2019s type inequalities for co-ordinated convex functions on quantum integral","volume":"20","author":"Alp","year":"2020","journal-title":"Appl. Math. -Notes"},{"key":"ref_17","first-page":"1","article-title":"On q-Hermite\u2013Hadamard inequalities for general convex functions","volume":"160","author":"Bermudo","year":"2020","journal-title":"Acta Math. Hung."},{"key":"ref_18","first-page":"207","article-title":"Quantum Hermite\u2013Hadamard type inequality and some estimates of quantum midpoint type inequalities for double integrals","volume":"37","author":"Kunt","year":"2019","journal-title":"Sigma J. Eng. Nat. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Wu, J.D., Hussain, S., and Latif, M.A. (2019). Simpson\u2019s type inequalities for co-ordinated convex functions on quantum calculus. Symmetry, 11.","DOI":"10.3390\/sym11060768"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.jksus.2016.07.001","article-title":"Some q-analogues of Hermite\u2013Hadamard inequality of functions of two variables on finite rectangles in the plane","volume":"29","author":"Latif","year":"2017","journal-title":"J. King-Saud Univ. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1016\/j.amc.2014.11.090","article-title":"Some quantum estimates for Hermite-Hadamard inequalities","volume":"251","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.amc.2015.07.078","article-title":"Some quantum integral inequalities via preinvex functions","volume":"269","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"781","DOI":"10.7153\/jmi-09-64","article-title":"Quantum integral inequalities for convex functions","volume":"9","author":"Sudsutad","year":"2015","journal-title":"J. Math. Inequal."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., 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_25","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., Liko, R., Kashuri, A., and Hern\u00e1ndez Hern\u00e1ndez, J.E. (2019). Quantum Estimates of Ostrowski Inequalities for Generalized \u03d5-Convex Functions. Symmetry, 11.","DOI":"10.3390\/sym11121513"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., Liko, R., Kashuri, A., and Hern\u00e1ndez Hern\u00e1ndez, J.E. (2020). Some New q-Integral Inequalities Using Generalized Quantum Montgomery Identity via Preinvex Functions. Symmetry, 12.","DOI":"10.3390\/sym12040553"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., Kashuri, A., Liko, R., and Hern\u00e1ndez Hern\u00e1ndez, J.E. (2020). Quantum Trapezium-Type Inequalities Using Generalized f-Convex Functions. Axioms, 9.","DOI":"10.3390\/axioms9010012"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"121","DOI":"10.18576\/amis\/130116","article-title":"Ostrowski-type inequalities for functions whose derivative modulus is relatively convex","volume":"13","author":"Cortez","year":"2019","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"931","DOI":"10.18576\/amis\/120505","article-title":"Jensen\u2019s inequality for convex functions on n-coordinates","volume":"12","author":"Viloria","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"79","DOI":"10.18576\/amis\/110110","article-title":"Ostrowski type inequalities for functions whose derivatives are (m,h1,h2)-convex","volume":"11","year":"2017","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.18576\/amis\/100507","article-title":"F\u00e9jer type inequalities for (s, m)-Convex functions in second sense","volume":"10","author":"Cortez","year":"2016","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Zhuang, H., Liu, W., and Park, J. (2019). Some quantum estimates of Hermite-Hadmard inequalities for quasi-convex functions. Mathematics, 7.","DOI":"10.3390\/math7020152"},{"key":"ref_33","first-page":"1","article-title":"Quantum integral inequalities on finite intervals","volume":"121","author":"Tariboon","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_34","first-page":"1","article-title":"Some new q-Hermite\u2013Hadamard like inequalities for co-ordinated convex functions","volume":"185","author":"Budak","year":"2020","journal-title":"J. Optim. Theory Appl."},{"key":"ref_35","unstructured":"Ali, M.A., Budak, H., and Sarikaya, M.Z. (2020, July 20). On Some New Trapezoidal Type Inequalities for the Functions of Two Variables via Quantum Calculus. Available online: https:\/\/www.researchgate.net\/publication\/342411316."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1476\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:08:06Z","timestamp":1760177286000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1476"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,8]]},"references-count":35,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091476"],"URL":"https:\/\/doi.org\/10.3390\/sym12091476","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,8]]}}}