{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:41:43Z","timestamp":1760139703742,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,24]],"date-time":"2022-04-24T00:00:00Z","timestamp":1650758400000},"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>In this investigation, we first establish new quantum Hermite\u2013Hadamard type integral inequalities for s-convex functions by utilizing newly defined Tq-integrals. Then, by using obtained inequality, we establish a new Hermite\u2013Hadamard inequality for coordinated s1,s2-convex functions. The results obtained in this paper provide significant extensions of other related results given in the literature. Finally, some examples are given to illustrate the result obtained in this paper. These types of analytical inequalities, as well as solutions, apply to different areas where the concept of symmetry is important.<\/jats:p>","DOI":"10.3390\/sym14050870","type":"journal-article","created":{"date-parts":[[2022,4,26]],"date-time":"2022-04-26T02:14:39Z","timestamp":1650939279000},"page":"870","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Some New Quantum Hermite-Hadamard Type Inequalities for s-Convex Functions"],"prefix":"10.3390","volume":"14","author":[{"given":"Ghazala","family":"Gulshan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mirpur University of Science and Technology (MUST), Mirpur 10250, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rashida","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mirpur University of Science and Technology (MUST), Mirpur 10250, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,24]]},"reference":[{"key":"ref_1","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2000). Selected topics on Hermite-Hadamard inequalities and applications. RGMIA Monographs, Victoria University."},{"key":"ref_2","unstructured":"Pe\u010dari\u0107, 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","first-page":"91","DOI":"10.1016\/S0893-9659(98)00086-X","article-title":"Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula","volume":"11","author":"Dragomi","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_4","first-page":"137","article-title":"Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula","volume":"147","author":"Kirmaci","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1016\/j.mcm.2011.12.048","article-title":"Hermite\u2013Hadamard\u2019s inequalities for fractional integrals and related fractional inequalities","volume":"57","author":"Sarikaya","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1080\/10652469.2013.824436","article-title":"On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals","volume":"25","year":"2014","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"775","DOI":"10.11650\/twjm\/1500574995","article-title":"On the Hadamard\u2019s inequality for functions on the co-ordinates in a rectangle from the plane","volume":"5","author":"Dragomir","year":"2001","journal-title":"Taiwan. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1063\/1.3525212","article-title":"Some Hadamard-Type inequalities for coordinated p-convex functions and Godunova-Levin functions","volume":"1309","author":"Akdemir","year":"2010","journal-title":"AIP Conf. Proc."},{"key":"ref_9","first-page":"1208","article-title":"Generalized inequalities of the type of Hermite-Hadamard-Fejer with quasi-convex functions by way of k-fractional derivative","volume":"22","author":"Ali","year":"2017","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_10","first-page":"1","article-title":"New quantum boundaries for quantum Simpson\u2019s and quantum Newton\u2019s type inequalities for preinvex functions","volume":"64","author":"Ali","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s41478-021-00323-8","article-title":"On some new trapezoidal inequalities for q\u03f02-quantum integrals via Green function","volume":"30","author":"Ali","year":"2021","journal-title":"J. Anal."},{"key":"ref_12","first-page":"14","article-title":"Inequalities of Hermite-Hadamard\u2019s type for functions whose derivatives absolute values are quasi-convex","volume":"12","author":"Alomari","year":"2009","journal-title":"Res. Rep. Coll."},{"key":"ref_13","first-page":"1","article-title":"An improvement of the Hermite-Hadamard inequality for functions convex on the coordinates","volume":"11","author":"Bakula","year":"2014","journal-title":"Aust. J. Math. Anal. Appl."},{"key":"ref_14","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_15","first-page":"23","article-title":"On some inequalities for product of different kinds of convex functions","volume":"5","year":"2020","journal-title":"Turk. J. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5529650","DOI":"10.1155\/2021\/5529650","article-title":"New post quantum analogues of Hermite\u2013Hadamard type inequalities for interval-valued convex functions","volume":"2021","author":"Kalsoom","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Rashid, S., Idrees, M., Safdar, F., Akram, S., Baleanu, D., and Chu, Y.M. (2020). Post quantum integral inequalities of Hermite-Hadamard-type associated with co-ordinated higher-order generalized strongly pre-invex and quasi-pre-invex mappings. Symmetry, 12.","DOI":"10.3390\/sym12030443"},{"key":"ref_18","first-page":"1","article-title":"Hermite-Hadamard type Inequalities for k-Riemann-Liouville fractional Integrals via two kinds of convexity","volume":"13","author":"Hussain","year":"2016","journal-title":"Austral. J. Math. Anal. Appl."},{"key":"ref_19","first-page":"301","article-title":"Some k-fractional associates of Hermite-Hadamard\u2019s inequality for quasi-convex functions and applications to special means","volume":"7","author":"Hussain","year":"2017","journal-title":"J. Fract. Differ. Calc."},{"key":"ref_20","first-page":"8","article-title":"Some generalized k-fractional companions of Hadamard\u2019s inequality","volume":"25","author":"Hussain","year":"2016","journal-title":"Niger. J. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.12785\/amis\/080318","article-title":"On the co-ordinated convex functions","volume":"8","author":"Ozdemir","year":"2014","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_22","first-page":"1","article-title":"New Refinements of Hadamard Integral inequlaity via k-Fractional Integrals for p-convex function","volume":"6","year":"2021","journal-title":"Turk. J. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1186\/s13662-014-0348-8","article-title":"New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations","volume":"2015","author":"Tariboon","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","first-page":"63","article-title":"Some Hadamard\u2019s inequalities for co-ordinated convex functions in a rectangle from the plane","volume":"11","author":"Wang","year":"2007","journal-title":"Taiwan. J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1515\/jaa-2014-0004","article-title":"Hermite-Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle","volume":"20","author":"Xi","year":"2014","journal-title":"J. Appl. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6634614","DOI":"10.1155\/2021\/6634614","article-title":"Quantum inequalities of Hermite-Hadamard type for r-convex functions","volume":"2021","author":"You","year":"2021","journal-title":"J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Zhuang, H., Liu, W., and Park, J. (2019). Some quantum estimates of Hermite-Hadamard inequalities for quasi-convex function. Mathematics, 7.","DOI":"10.3390\/math7020152"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/BF01837981","article-title":"Some remarks on s-convex functions","volume":"48","author":"Hudzik","year":"1994","journal-title":"Aequationes Math."},{"key":"ref_29","first-page":"686","article-title":"The Hadamard inequalities for s-convex functions in the second sense","volume":"32","author":"Dragomir","year":"1999","journal-title":"Demonstr. Math."},{"key":"ref_30","first-page":"629","article-title":"The Hadamards inequality for s-convex function of 2-variables on the coordinates","volume":"2","author":"Alomari","year":"2008","journal-title":"Int. J. Math. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1643","DOI":"10.1016\/S0252-9602(11)60350-0","article-title":"Some inequalities of Hermite-Hadamard type for s-convex functions","volume":"31","author":"Alomari","year":"2011","journal-title":"Acta Math. Sci."},{"key":"ref_32","first-page":"5171","article-title":"New inequalities of Hermite\u2013Hadamard type via s-convex functions in the second sense with applications","volume":"17","author":"Avci","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"705","DOI":"10.22436\/jnsa.009.02.32","article-title":"Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions","volume":"9","author":"Chen","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1414","DOI":"10.1515\/math-2017-0121","article-title":"Some new inequalities of Hermite-Hadamard type for s-convex functions with applications","volume":"15","author":"Khan","year":"2017","journal-title":"Open Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s00010-019-00642-z","article-title":"An extension of the Hermite\u2013Hadamard inequality for convex and s-convex functions","volume":"93","year":"2019","journal-title":"Aequationes Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"491","DOI":"10.18514\/MMN.2015.1099","article-title":"Some new inequalities of Hermite-Hadamard type for s-convex functions","volume":"16","author":"Sarikaya","year":"2015","journal-title":"Miskolc Math. Notes"},{"key":"ref_37","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_38","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2001). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_39","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_40","doi-asserted-by":"crossref","first-page":"13327","DOI":"10.3934\/math.2021771","article-title":"Quantum Hermite-Hadamard and quantum Ostrowski type inequalities for s-convex functions in the second sense with applications","volume":"6","author":"Asawasamrit","year":"2021","journal-title":"AIMS Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2035","DOI":"10.26637\/MJM0804\/0121","article-title":"q-Inequalities on quantum integral","volume":"8","author":"Alp","year":"2020","journal-title":"Malaya J. Mat."},{"key":"ref_42","first-page":"146","article-title":"A new Definition and properties of quantum integral which calls q\u00af-integral","volume":"5","author":"Alp","year":"2017","journal-title":"Konuralp J. Math."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1186\/s13660-021-02715-7","article-title":"On new generalized quantum integrals and related Hermite-Hadamard inequalities","volume":"2021","author":"Kara","year":"2021","journal-title":"J. Inequalities Appl."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Kara, H., and Budak, H. (2021). On Hermite-Hadamard type inequalities for newly defined generalized quantum integrals. Ric. Mat.","DOI":"10.1186\/s13660-021-02715-7"},{"key":"ref_45","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."},{"key":"ref_46","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. Hungar."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.jksus.2016.07.001","article-title":"Some q-analogues of Hermite-Hadamard inequality of functions of two variables on finite rectangles in the plane","volume":"29","author":"Latif","year":"2017","journal-title":"J. King Saud Univ."},{"key":"ref_48","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_49","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."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/870\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:59:45Z","timestamp":1760137185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/870"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,24]]},"references-count":49,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050870"],"URL":"https:\/\/doi.org\/10.3390\/sym14050870","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,4,24]]}}}