{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:34:27Z","timestamp":1767918867900,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T00:00:00Z","timestamp":1672617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok","award":["5542101"],"award-info":[{"award-number":["5542101"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q-differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities.<\/jats:p>","DOI":"10.3390\/axioms12010049","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T04:56:27Z","timestamp":1672635387000},"page":"49","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["On New Estimates of q-Hermite\u2013Hadamard Inequalities with Applications in Quantum Calculus"],"prefix":"10.3390","volume":"12","author":[{"given":"Saowaluck","family":"Chasreechai","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5341-4926","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":"Muhammad Amir","family":"Ashraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Agriculture Faisalabad, Faisalabad 38000, Pakistan"}]},{"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1574-1800","authenticated-orcid":false,"given":"Sina","family":"Etemad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 3751-71379, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel De la","family":"Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV\/EHU), 48940 Leioa, Bizkaia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3463-2607","authenticated-orcid":false,"given":"Shahram","family":"Rezapour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 3751-71379, Iran"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"9479","DOI":"10.3934\/math.2022526","article-title":"On degree theory for non-monotone type fractional order delay differential equations","volume":"7","author":"Shah","year":"2022","journal-title":"AIMS Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ahmad, S.W., Sarwar, M., Shah, K., and Abdeljawad, T. (2022). Study of a coupled system with sub-strip and multi-valued boundary conditions via topological degree theory on an infinite domain. Symmetry, 14.","DOI":"10.3390\/sym14050841"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"111955","DOI":"10.1016\/j.chaos.2022.111955","article-title":"Computational study on the dynamics of fractional order differential equations with applications","volume":"157","author":"Shah","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_4","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1908","journal-title":"Trans. R. Soc. Edinb."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ahmad, B., Ntouyas, S.K., and Tariboon, J. (2016). Quantum Calculus: New Concepts, Impulsive IVPs and BVPs, Inequalities, World Scientific.","DOI":"10.1142\/10075"},{"key":"ref_7","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_8","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2001). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s10958-021-05568-z","article-title":"Oscillation and nonoscillation results for the Caputo fractional q-difference equations and inclusions","volume":"258","author":"Abbas","year":"2021","journal-title":"J. Math. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1186\/s13661-018-0962-6","article-title":"On four-point fractional q-integro-difference boundary value problems involving separate nonlinearity and arbitrary fraction order","volume":"2018","author":"Patanarapeelert","year":"2018","journal-title":"Bound. Value Probl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1186\/s13662-021-03652-x","article-title":"Positive solutions for eigenvalue problems of fractional q-difference equation with \u03d5-Laplacian","volume":"2021","author":"Wang","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ouncharoen, R., Patanarapeelert, N., and Sitthiwirattham, T. (2018). Nonlocal q-symmetric integral boundary value problem for sequential q-symmetric integro-difference equations. Mathematics, 6.","DOI":"10.3390\/math6110218"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hajiseyedazizi, S.N., Samei, M.E., Alzabut, J., and Chu, Y.M. (2021). On multi-step methods for singular fractional q-integro-differential equations. Open Math., 19.","DOI":"10.1515\/math-2021-0093"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Etemad, S., Ntouyas, S.K., and Ahmad, B. (2019). Existence theory for a fractional q-integro-difference equation with q-integral boundary conditions of different orders. Mathematics, 7.","DOI":"10.3390\/math7080659"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1186\/s13660-019-2257-6","article-title":"Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach","volume":"2019","author":"Butt","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1186\/s13662-020-02766-y","article-title":"Solutions of two fractional q-integro-differential equations under sum and integral boundary value conditions on a time scale","volume":"2020","author":"Alzabut","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Rezapour, S., Imran, A., Hussain, A., Martinez, F., Etemad, S., and Kaabar, M.K.A. (2021). Condensing functions and approximate endpoint criterion for the existence analysis of quantum integro-difference FBVPs. Symmetry, 13.","DOI":"10.3390\/sym13030469"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1186\/s13662-021-03525-3","article-title":"On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria","volume":"2021","author":"Boutiara","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1186\/s13662-020-03092-z","article-title":"A novel fractional structure of a multi-order quantum multi-integro-differential problem","volume":"2020","author":"Phuong","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_20","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2000). Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University."},{"key":"ref_21","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_22","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1007\/s10474-020-01025-6","article-title":"On q-Hermite-Hadamard inequalities for general convex functions","volume":"162","author":"Bermudo","year":"2020","journal-title":"Acta Math. Hung."},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"1","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_25","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_26","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_27","doi-asserted-by":"crossref","unstructured":"Ding, Y., Kalsoom, H., and Wu, S. (2019). Some new quantum Hermite\u2013Hadamard-type estimates within a class of generalized (s,m)-preinvex functions. Symmetry, 11.","DOI":"10.3390\/sym11101283"},{"key":"ref_28","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_29","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_30","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_31","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_32","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_33","doi-asserted-by":"crossref","first-page":"1","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_34","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_35","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_36","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_37","doi-asserted-by":"crossref","unstructured":"Budak, H., Ali, M.A., Alp, N., and Chu, Y.-M. (J. Math. Inequal., 2021). Quantum Ostrowski type integral inequalities, J. Math. Inequal., in press.","DOI":"10.1002\/mma.7153"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","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_39","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2001). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Sial, I.B., Mei, S., Ali, M.A., and Nanlaopon, K. (2021). On some generalized Simpson\u2019s and Newton\u2019s inequalities for (\u03b1,m)-convex functions in q-calculus. Mathematics, 2021.","DOI":"10.3390\/math9243266"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Soontharanon, J., Ali, M.A., Budak, H., Nanlaopon, K., and Abdullah, Z. (2022). Simpson\u2019s and Newton\u2019s Inequalities for (\u03b1,m)-Convex Functions via Quantum Calculus. Symmetry, 14.","DOI":"10.3390\/sym14040736"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:56:18Z","timestamp":1760118978000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,2]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010049"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010049","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,2]]}}}