{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T23:09:56Z","timestamp":1775948996890,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,6,29]],"date-time":"2021-06-29T00:00:00Z","timestamp":1624924800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11971241"],"award-info":[{"award-number":["11971241"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this investigation, for convex functions, some new (p,q)\u2013Hermite\u2013Hadamard-type inequalities using the notions of (p,q)\u03c02 derivative and (p,q)\u03c02 integral are obtained. Furthermore, for (p,q)\u03c02-differentiable convex functions, some new (p,q) estimates for midpoint and trapezoidal-type inequalities using the notions of (p,q)\u03c02 integral are offered. It is also shown that the newly proved results for p=1 and q\u21921\u2212 can be converted into some existing results. Finally, we discuss how the special means can be used to address newly discovered inequalities.<\/jats:p>","DOI":"10.3390\/e23070828","type":"journal-article","created":{"date-parts":[[2021,6,29]],"date-time":"2021-06-29T10:52:46Z","timestamp":1624963966000},"page":"828","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["Some New Hermite\u2013Hadamard and Related Inequalities for Convex Functions via (p,q)-Integral"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"first","affiliation":[{"name":"Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, University of Buenos Aires, Av. 12 de octubre 1076 y Roca, Apartado Postal 17-01-2184, Sede Quito, Ecuador"}]},{"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8843-955X","authenticated-orcid":false,"given":"H\u00fcseyin","family":"Budak","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, D\u00fczce University, D\u00fczce 81620, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-3349","authenticated-orcid":false,"given":"Humaira","family":"Kalsoom","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, 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 303012, India"},{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates"},{"name":"International Center for Basic and Applied Sciences, Jaipur 302029, India"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,29]]},"reference":[{"key":"ref_1","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2000). Selected Topics on Hermite-Hadamard Inequalities and Applications, Victoria University. RGMIA Monographs."},{"key":"ref_2","unstructured":"Pe\u0107arixcx, J.E., Proschan, F., and Tong, Y.L. (1992). Convex Functions, Partial Orderings and Statistical Applications, Academic Press."},{"key":"ref_3","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_4","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2001). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_5","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_6","doi-asserted-by":"crossref","unstructured":"Bokulich, A., and Jaeger, G. (2010). Philosophy of Quantum Information Theory and Entaglement, Cambridge Uniersity Press.","DOI":"10.1017\/CBO9780511676550"},{"key":"ref_7","unstructured":"Ernst, T. (2000). The History Of Q-Calculus Furthermore, New Method, Department of Mathematics, Uppsala University."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1017\/S0013091500011469","article-title":"Some fractional q-integrals and q-derivatives","volume":"15","year":"1966","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_10","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_11","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_12","unstructured":"Sadjang, P.N. (2013). On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas. arXiv."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","unstructured":"Chu, Y.-M., Awan, M.U., Talib, S., Noor, M.A., and Noor, K.I. (2020). New post quantum analogues of Ostrowski-type inequalities using new definitions of left\u2013right (p,q)-derivatives and definite integrals. Adv. Differ. Equ., 634.","DOI":"10.1186\/s13662-020-03094-x"},{"key":"ref_15","first-page":"199","article-title":"Some trapezoide and midpoint type inequalities for newly defined quantum integrals","volume":"40","author":"Budak","year":"2021","journal-title":"Proycciones J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ali, M.A., Budak, H., Abbas, M., and Chu, Y.-M. (2021). Quantum Hermite\u2013Hadamard-type inequalities for functions with convex absolute values of second qb-derivatives. Adv. Differ. Equ., 7.","DOI":"10.1186\/s13662-020-03163-1"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ali, M.A., Alp, N., Budak, H., Chu, Y.-M., and Zhang, Z. (2021). On some new quantum midpoint type inequalities for twice quantum differentiable convex functions. Open Math., in press.","DOI":"10.22541\/au.161400461.14533814\/v1"},{"key":"ref_18","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_19","first-page":"341","article-title":"Hermite Hadamard\u2019s Type Inequalities for Co-Ordinated Convex Functions on Quantum Integral","volume":"20","author":"Alp","year":"2020","journal-title":"Appl. Math. E-Notes"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s10957-020-01726-6","article-title":"Some New Quantum Hermite-Hadamard-Like Inequalities for Coordinated Convex Functions","volume":"186","author":"Budak","year":"2020","journal-title":"J. Optim. Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Jain, S., Mehrez, K., Baleanu, D., and Agarwal, P. (2019). Certain Hermite\u2013Hadamard inequalities for logarithmically convex functions with applications. Mathematics, 7.","DOI":"10.3390\/math7020163"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Jhanthanam, S., Tariboon, J., Ntouyas, S.K., and Nonlaopon, K. (2019). On q-Hermite-Hadamard Inequalities for Differentiable Convex Functions. Mathematics, 7.","DOI":"10.3390\/math7070632"},{"key":"ref_23","first-page":"501","article-title":"Some quantum estimates of Hermite-Hadamard inequalities for convex functions","volume":"7","author":"Liu","year":"2016","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_24","first-page":"675","article-title":"Some quantum estimates for Hermite-Hadamard inequalities","volume":"251","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"101","DOI":"10.5269\/bspm.v38i1.32820","article-title":"Hermite-Hadamard type inequalities for generalized convex functions on fractal sets style","volume":"38","author":"Tomar","year":"2020","journal-title":"Bol. Soc. Parana. Matem\u00e1Tica"},{"key":"ref_26","first-page":"242","article-title":"Some quantum integral inequalities via preinvex functions","volume":"269","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_27","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_28","first-page":"1","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_29","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":"2021","journal-title":"Math. Methods Appl. Sci."},{"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. Methods Appl. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1186\/s13662-021-03226-x","article-title":"New quantum boundaries for quantum Simpson\u2019s and quantum Newton\u2019s type inequalities for preinvex functions","volume":"2021","author":"Ali","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., Ali, M.A., Kashuri, A., Sial, I.B., and Zhang, Z. (2020). Some New Newton\u2019s Type Integral Inequalities for Co-Ordinated Convex Functions in Quantum Calculus. Symmetry, 12.","DOI":"10.3390\/sym12091476"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","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_34","doi-asserted-by":"crossref","unstructured":"Ali, M.A., Budak, H., Akkurt, A., and Chu, Y.-M. (2021). Quantum Ostrowski type inequalities for twice quantum differentiable functions in quantum calculus. Open Math., in press.","DOI":"10.1515\/math-2021-0020"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Budak, H., Ali, M.A., Alp, N., and Chu, Y.-M. (2021). Quantum Ostrowski type integral inequalities. J. Math. Inequalities, in press.","DOI":"10.1002\/mma.7153"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s13398-017-0402-y","article-title":"(p,q)-Hermite-Hadamard 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 y Nat. Ser. A Matem\u00e1ticas"},{"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","unstructured":"Roberts, A.W., and Varberg, D.E. (1973). Convex Functions, Academic Press."},{"key":"ref_39","first-page":"7","article-title":"Convex functions and the Hadamard inequality","volume":"28","author":"Azpetitia","year":"1994","journal-title":"Rev. Colomb. Mat."},{"key":"ref_40","first-page":"137","article-title":"Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula","volume":"147","author":"Kirmaci","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0893-9659(98)00086-X","article-title":"Two inequalities for diferentiable mappings and applications to special means fo real numbers and to trapezoidal formula","volume":"11","author":"Dragomir","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0893-9659(99)00164-0","article-title":"Inequalities for diferentiable mappings with application to special means and quadrature formulae","volume":"13","author":"Pearce","year":"2000","journal-title":"Appl. Math. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/7\/828\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:26:42Z","timestamp":1760164002000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/7\/828"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,29]]},"references-count":42,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["e23070828"],"URL":"https:\/\/doi.org\/10.3390\/e23070828","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,29]]}}}