{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:02:56Z","timestamp":1760234576503,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T00:00:00Z","timestamp":1622073600000},"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 paper, we use (p,q)-integral to establish some Fej\u00e9r type inequalities. In particular, we generalize and correct existing results of quantum Fej\u00e9r type inequalities by using new techniques and showing some problematic parts of those results. Most of the inequalities presented in this paper are significant extensions of results which appear in existing literatures.<\/jats:p>","DOI":"10.3390\/sym13060953","type":"journal-article","created":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T11:07:02Z","timestamp":1622113622000},"page":"953","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On Fej\u00e9r Type Inequalities via (p,q)-Calculus"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4618-8585","authenticated-orcid":false,"given":"Nuttapong","family":"Arunrat","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7550-7744","authenticated-orcid":false,"given":"Keaitsuda Maneeruk","family":"Nakprasit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8185-3539","authenticated-orcid":false,"given":"Jessada","family":"Tariboon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,27]]},"reference":[{"key":"ref_1","first-page":"193","article-title":"On a q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Quart. J. Pure Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/2370183","article-title":"q-difference equations","volume":"32","author":"Jackson","year":"1910","journal-title":"Am. J. Math."},{"key":"ref_3","first-page":"1013","article-title":"Quantum Ostrowski inequalities for q-differentiable convex function","volume":"10","author":"Aslam","year":"2016","journal-title":"J. Math. Inequal."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Aral, A., Gupta, V., and Agarwal, R.P. (2013). Applications of q-Calculus in Operator Theory, Springer Science + Business Media.","DOI":"10.1007\/978-1-4614-6946-9"},{"key":"ref_5","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":"2002","journal-title":"J. Comput. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2011\/107384","article-title":"Boundary-value problems for nonlinear third-order q-difference equations","volume":"94","author":"Ahmad","year":"2011","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1186\/1687-1847-2012-35","article-title":"A study of second-order q-difference equations with boundary conditions","volume":"2012","author":"Ahmad","year":"2012","journal-title":"Adv. Differ. Equ."},{"key":"ref_8","first-page":"59","article-title":"Existence results for nonlinear q-difference equations with nonlocal boundary conditions","volume":"19","author":"Ahmad","year":"2012","journal-title":"Commun. Appl. Nonlinear Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1186\/1687-1847-2012-81","article-title":"On nonlocal boundary value problems of nonlinear q-difference equation","volume":"2012","author":"Ahmad","year":"2012","journal-title":"Adv. Differ. Equ."},{"key":"ref_10","unstructured":"Bukweli-Kyemba, J.D., and Hounkonnou, M.N. (2013). Quantum deformed algebra: Coherent states and special functions. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2002). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"282","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_13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1186\/1029-242X-2014-121","article-title":"Quantum integral inequalities on finite intervals","volume":"2014","author":"Tariboon","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_14","first-page":"207","article-title":"Quantum Hermite-Hadamard 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_15","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_16","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_17","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_18","doi-asserted-by":"crossref","first-page":"675","DOI":"10.7153\/jmi-2019-13-45","article-title":"Quantum Hermite-Hadamard inequalities for double integral and q-differentiable convex functions","volume":"13","author":"Prabseang","year":"2019","journal-title":"J. Math. Inequal."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"875","DOI":"10.7153\/jmi-2020-14-57","article-title":"On refinement of quantum Hermite-Hadamard inequalities for convex functions","volume":"14","author":"Prabseang","year":"2020","journal-title":"J. Math. Inequal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"L711","DOI":"10.1088\/0305-4470\/24\/13\/002","article-title":"A (p,q)-oscillator realization of two-parameter quantum algebras","volume":"24","author":"Chakrabarti","year":"1991","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_22","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_23","first-page":"1","article-title":"(p,q)-integral inequalities","volume":"19","year":"2016","journal-title":"RGMIA Res. Rep. Coll"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Prabseang, J., Nonlaopon, K., and Tariboon, J. (2019). (p,q)-Hermite-Hadamard inequalities for double integral and (p,q)-differentiable convex functions. Axioms, 8.","DOI":"10.3390\/axioms8020068"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Amer, M., Junjua, M.D., Hassain, S., and Shahxadi, G. (2019). (p,q)-estimates of Hermite-Hadamard-type inequalities for coordinated convex and quasi convex function. Mathematics, 7.","DOI":"10.3390\/math7080683"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"145","DOI":"10.55937\/sut\/1394548362","article-title":"(p,q)-calculus: Differentiation and integration","volume":"49","author":"Hounkonnou","year":"2013","journal-title":"SUT J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s00025-018-0783-z","article-title":"On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas","volume":"73","author":"Sadjang","year":"2018","journal-title":"Results Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1186\/s13662-020-03094-x","article-title":"New post quantum analogues of Ostrowski-type inequalities using new definitions of left-right (p,q)-derivatives and definite integrals","volume":"2020","author":"Chu","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Rashid, S., Tdrees, M., Safdar, F., Akram, S., Baleanu, D., and Chu, Y.M. (2020). Post quantum inequalities of Hermite-Hadamard-type associated with co-ordinated higher-order generalized strongly pre-index and quasi-pre-index mappings. Symmetry, 12.","DOI":"10.3390\/sym12030443"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s13398-017-0402-y","article-title":"(p,q)-Hermite-Hadamard and (p,q)-estimates for midpoint type inequalities via convex and quasi-convex functions","volume":"112","author":"Kunt","year":"2018","journal-title":"RACSAM"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ali, M.A., Budak, H., Kalsoom, H., and Chu, Y.M. (2020). Post-quantum Hermite-Hadamard inequalities involving newly defined (p,q)-integral. Authorea.","DOI":"10.22541\/au.160465507.75463188\/v1"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"77","DOI":"10.3934\/math.2021006","article-title":"Some (p,q)-Hardy type inequalities for (p,q)-integrable functions","volume":"6","author":"Thongjob","year":"2020","journal-title":"AIMS Math."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Wannalookkhee, F.S., Nonlaopon, K., Tariboon, J., and Ntouyas, S.K. (2021). On Hermite-Hadamard type inequalities for coordinated convex functions via (p,q)-calculus. Mathematics, 9.","DOI":"10.22541\/au.163332914.42587950\/v1"},{"key":"ref_34","unstructured":"Pe\u010dari\u0107, J., Proschan, F., and Tong, Y.L. (1991). Convex Functions, Partial Ordering and Statistical Applications, Academic Press."},{"key":"ref_35","first-page":"63","article-title":"Some remarks on the midpoint rule in numerical integration","volume":"XLV","author":"Dragomir","year":"2000","journal-title":"Studia Univ. Babes. Bolyai. Math."},{"key":"ref_36","first-page":"475","article-title":"Some remarks on the trapezoid rule in numerical integration","volume":"31","author":"Dragomir","year":"2000","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_37","first-page":"377","article-title":"On new Hermite Hadamard Fej\u00e9r type integral inequalities","volume":"57","author":"Sarikaya","year":"2012","journal-title":"Studia Univ. Babes. Bolyai. Math."},{"key":"ref_38","first-page":"1","article-title":"Generalized Ostrowski type inequalities for local fractional integrals","volume":"62","author":"Sarikaya","year":"2015","journal-title":"RGMIA Res. Rep. Collect."},{"key":"ref_39","first-page":"1","article-title":"Some generalized Ostrowski type inequalities involving local fractional integrals and applications","volume":"2016","author":"Sarikaya","year":"2016","journal-title":"Adv. Inequal. Appl."},{"key":"ref_40","first-page":"1","article-title":"On generalized Hermite-Hadamard inequality for generalized convex function","volume":"64","author":"Sarikaya","year":"2015","journal-title":"RGMIA Res. Rep. Collect."},{"key":"ref_41","first-page":"1","article-title":"Some integral inequalities for local fractional integrals","volume":"65","author":"Sarikaya","year":"2015","journal-title":"RGMIA Res. Rep. Collect."},{"key":"ref_42","first-page":"1","article-title":"On new inequalities of Simpson\u2019s type for generalized convex functions","volume":"66","author":"Sarikaya","year":"2015","journal-title":"RGMIA Res. Rep. Collect."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.11650\/twjm\/1500406376","article-title":"Some inequalities for differentiable mappings and applications to Feje\u00b4r inequality and weighted trapozidal formula","volume":"15","author":"Tseng","year":"2011","journal-title":"Taiwan. J. Math."},{"key":"ref_44","first-page":"567","article-title":"On an extension of Hadamard inequality for convex functions","volume":"3","author":"Wang","year":"1982","journal-title":"Chin. Ann. Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1741","DOI":"10.1216\/RMJ-2009-39-5-1741","article-title":"On the weighted generalization of the Hermite-Hadamard inequality and its applications","volume":"39","author":"Wu","year":"2009","journal-title":"Rocky Mt. J. Math."},{"key":"ref_46","first-page":"243","article-title":"Some Hermite-Hadamard type inequalities for differentiable convex functions and applications","volume":"42","author":"Xi","year":"2013","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_47","first-page":"163","article-title":"Hermite-Hadamard type inequalities for functions whose derivatives are of convexities","volume":"18","author":"Xi","year":"2013","journal-title":"Nonlinear Funct. Anal. Appl."},{"key":"ref_48","first-page":"369","article-title":"Uber die Fourierreihen","volume":"24","year":"1906","journal-title":"II. Math. Naturwiss. Anz Ungar. Akad. Wiss."},{"key":"ref_49","first-page":"1","article-title":"Fej\u00e9r-type inequalities","volume":"9","author":"Minculete","year":"2012","journal-title":"Aust. J. Math. Anal. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"123","DOI":"10.2306\/scienceasia1513-1874.2017.43.123","article-title":"Some new Fej\u00e9r type inequalities via quantum calculus on finite intervals","volume":"2017","author":"Yang","year":"2017","journal-title":"Sci. Asia"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1080\/09720502.2015.1023545","article-title":"On Fej\u00e9r type inequalities via fractional integrals","volume":"21","author":"Sarikaya","year":"2018","journal-title":"J. Interdiscip. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/953\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:09:03Z","timestamp":1760162943000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/953"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,27]]},"references-count":51,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13060953"],"URL":"https:\/\/doi.org\/10.3390\/sym13060953","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,5,27]]}}}