{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:51:17Z","timestamp":1760241077771,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,11,13]],"date-time":"2019-11-13T00:00:00Z","timestamp":1573603200000},"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 the present work, the Hermite\u2013Hadamard inequality is established in the setting of quantum calculus for a generalized class of convex functions depending on three parameters: a number in     ( 0 , 1 ]     and two arbitrary real functions defined on     [ 0 , 1 ]    . From the proven results, various inequalities of the same type are deduced for other types of generalized convex functions and the methodology used reveals, in a sense, a symmetric mathematical phenomenon. In addition, the definition of dominated convex functions with respect to the generalized class of convex functions aforementioned is introduced, and some integral inequalities are established.<\/jats:p>","DOI":"10.3390\/sym11111402","type":"journal-article","created":{"date-parts":[[2019,11,13]],"date-time":"2019-11-13T09:11:27Z","timestamp":1573636287000},"page":"1402","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Some Inequalities Using Generalized Convex Functions in Quantum Analysis"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel J.","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. Apartado, Quito 17-01-2184, Ecuador"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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, L. Pavaresia, Vlora 1001, Vlore, Albania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2439-8538","authenticated-orcid":false,"given":"Rozana","family":"Liko","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, L. Pavaresia, Vlora 1001, Vlore, Albania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4406-5469","authenticated-orcid":false,"given":"Jorge E. Hern\u00e1ndez","family":"Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Departamento de T\u00e9cnicas Cuantitativas, Decanato de Ciencias Econ\u00f3micas y Empresariales, Universidad Centroccidental Lisandro Alvarado, Av. 20. esq. Av. Moran, Edf. Los Militares, Piso 2, Ofc.2, Barquisimeto 3001, Venezuela"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,13]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"81","DOI":"10.3176\/proc.2008.2.03","article-title":"The different tongues of q-calculus","volume":"57","author":"Ernst","year":"2008","journal-title":"Proc. Eston. Acad. Sci."},{"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","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_5","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2002). Quantun Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ismail, M.E.H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press.","DOI":"10.1017\/CBO9781107325982"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1016\/j.aam.2010.02.003","article-title":"q-analogues of Freud weights and nonlinear difference equations","volume":"45","author":"Ismail","year":"2010","journal-title":"Adv. Appl. Math."},{"key":"ref_8","first-page":"194","article-title":"Companion Inequalities to Jensen\u2019s Inequality for m-convex and s,m-Convex Functions","volume":"7","author":"Bakula","year":"2006","journal-title":"J. Ineq. Pure Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"625","DOI":"10.3906\/mat-1604-45","article-title":"The most important inequalities for m-convex functions","volume":"41","author":"Ardic","year":"2017","journal-title":"Turk. J. Math."},{"key":"ref_10","first-page":"1557","article-title":"Co-ordinated s-Convex Function in the First Sense with Some Hadamard-Type Inequalities","volume":"3","author":"Alomari","year":"2008","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_11","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":"31B","author":"Alomari","year":"2011","journal-title":"Acta Math. Sci."},{"key":"ref_12","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":"139","author":"Akdemir","year":"2010","journal-title":"AIP Conf. Proc."},{"key":"ref_13","unstructured":"Mihe\u015fan, V.G. (,  1993). A generalization of the convexity. Proceedings of the Seminar on Functional Equations, Approximation and Convexity, Cluj-Napoca, Romania."},{"key":"ref_14","first-page":"327","article-title":"A Generalization of Simpson Type Inequality via Differentiable function using (s,m)-convex functions","volume":"35","author":"Yang","year":"2015","journal-title":"Ital. J. Pur. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"489","DOI":"10.7153\/jmi-08-36","article-title":"Some Remarks on s,m-convexity in the second sense","volume":"8","author":"Eftekhari","year":"2014","journal-title":"J. Math. Ineq."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"766","DOI":"10.22436\/jnsa.009.03.05","article-title":"Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals","volume":"9","author":"Liu","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_17","first-page":"679","article-title":"On tgs-convex function and their inequalities","volume":"30","author":"Gov","year":"2015","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_18","first-page":"23","article-title":"Hermite-Hadamard Type Inequalities for (m,h1,h2)-Convex Functions Via Riemann-Liouville Fractional","volume":"2","author":"Shi","year":"2014","journal-title":"Integr. Turk. J. Anal. Number Theory"},{"key":"ref_19","first-page":"33","article-title":"Upper and Lower Bounds for Riemann Type Quantum Integrals of Preinvex and Preinvex dominates functions","volume":"79","author":"Awan","year":"2017","journal-title":"UPB Sci. Bull. Ser. A"},{"key":"ref_20","first-page":"1","article-title":"Quantum Integral Inequalities via \u03c6-Convex Functions","volume":"4","author":"Noor","year":"2016","journal-title":"Quant. Inf. Rev."},{"key":"ref_21","first-page":"13","article-title":"Stetigkeitsaussagen f\u00fcr eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen R\u00e4aumen","volume":"23","author":"Breckner","year":"1978","journal-title":"Pub. Inst. Math."},{"key":"ref_22","first-page":"67","article-title":"New Inequalities of Ostrowski\u2019s Type for s-Convex Functions in the Second Sense with Applications","volume":"27","author":"Set","year":"2012","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"107","DOI":"10.4064\/sm-21-1-107-115","article-title":"A note on the theory of s-normed spaces of \u03c6-integrable functions","volume":"21","author":"Matuszewska","year":"1961","journal-title":"Studia Math."},{"key":"ref_24","first-page":"157","article-title":"A note on modular spaces I","volume":"9","author":"Orlicz","year":"1951","journal-title":"Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys."},{"key":"ref_25","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":"Aequ. Math."},{"key":"ref_26","first-page":"335","article-title":"Some Inequalities of Hadamard Type","volume":"21","author":"Dragomir","year":"1995","journal-title":"Soochow J. Math."},{"key":"ref_27","first-page":"21","article-title":"On some inequalities for convex\u2013dominated functions","volume":"19","author":"Dragomir","year":"1990","journal-title":"Anal. Num. Theor. Approx."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"173","DOI":"10.37193\/CJM.2015.02.04","article-title":"Bounds of the second degree cumulative frontier gaps of functions with generalized convexity","volume":"31","author":"Cristescu","year":"2015","journal-title":"Carpath. J. Math."},{"key":"ref_29","first-page":"687","article-title":"The Hadamard\u2019s inequality for s -convex functions in the second sense","volume":"32","author":"Dragomir","year":"1999","journal-title":"Demonstr. Math."},{"key":"ref_30","first-page":"135","article-title":"Some new quantum inequalities via tgs-convex functions","volume":"9","author":"Noor","year":"2018","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_31","first-page":"161","article-title":"Means, g-Convex Dominated & Hadamard-Type Inequalities","volume":"18","author":"Dragomir","year":"2002","journal-title":"Tam. Oxf. J. Math. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1402\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:33:58Z","timestamp":1760189638000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/11\/1402"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,13]]},"references-count":31,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["sym11111402"],"URL":"https:\/\/doi.org\/10.3390\/sym11111402","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,11,13]]}}}