{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T20:56:43Z","timestamp":1775595403099,"version":"3.50.1"},"reference-count":74,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T00:00:00Z","timestamp":1616112000000},"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>Fractional q-calculus has been investigated and applied in a variety of fields in mathematical areas including fractional q-integral inequalities. In this paper, we study fractional (p,q)-calculus on finite intervals and give some basic properties. In particular, some fractional (p,q)-integral inequalities on finite intervals are proven.<\/jats:p>","DOI":"10.3390\/sym13030504","type":"journal-article","created":{"date-parts":[[2021,3,21]],"date-time":"2021-03-21T23:47:41Z","timestamp":1616370461000},"page":"504","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Fractional (p,q)-Calculus on Finite Intervals and Some Integral Inequalities"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1323-9558","authenticated-orcid":false,"given":"Pheak","family":"Neang","sequence":"first","affiliation":[{"name":"Department of Mathematics, 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, 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. 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Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1186\/s13660-019-2101-z","article-title":"Some results on quantum Hanh integral inequalities","volume":"2019","author":"Asawasamrit","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.jmaa.2004.12.029","article-title":"Variational calculus on q-nonuniform","volume":"306","author":"Bangerezako","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","unstructured":"Exton, H. (1983). q-Hypergeometric Functions and Applications, Hastead Press."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Annyby, H.M., and Mansour, S.K. (2012). q-Fractional Calculus and Equations, Springer.","DOI":"10.1007\/978-3-642-30898-7"},{"key":"ref_9","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_10","unstructured":"Ernst, T. (2000). A History of q-Calculus and a New Method, Department of Mathematics, Uppsala University. UUDM Report 2000:16."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2010.1.70","article-title":"Nontrivial solutions for fractional q-difference boundary value problems","volume":"2010","author":"Ferreira","year":"2010","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.7153\/jmi-10-81","article-title":"Quantum Ostrowski inequalities for q-differentiable convex function","volume":"10","author":"Noor","year":"2016","journal-title":"J. Math. Inequal."},{"key":"ref_13","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_14","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_15","first-page":"122","article-title":"Riemann\u2013Liouville fractional quantum Hermite\u2013Hadamard type inequalities for convex functions","volume":"8","author":"Kunt","year":"2020","journal-title":"Konuralp J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/j.cam.2005.06.009","article-title":"A second order q-difference equation solvable by factorization method","volume":"193","author":"Dobrogowska","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s11139-006-0259-8","article-title":"Some systems of multivariable orthogonal q-Racah polynomials","volume":"13","author":"Gasper","year":"2007","journal-title":"Ramanujan J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1016\/j.cam.2009.02.044","article-title":"q-Difference operators for orthogonal polynomials","volume":"233","author":"Ismail","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","first-page":"75","article-title":"The h-Laplace and q-Laplace transforms","volume":"365","author":"Bohner","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","first-page":"1733","article-title":"Positive solutions of q-difference equation","volume":"138","author":"Hassan","year":"2010","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_21","first-page":"94","article-title":"Boundary-value problems for nonlinear third-order q-difference equations","volume":"2011","author":"Ahmad","year":"2011","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_22","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_23","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_24","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_25","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2002). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_26","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_27","first-page":"121","article-title":"Quantum integral inequalities on finite intervals","volume":"2014","author":"Tariboon","year":"2014","journal-title":"Adv. Differ. Equ."},{"key":"ref_28","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 functions and quasi-convex functions","volume":"30","author":"Necmettin","year":"2018","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_29","first-page":"781","article-title":"Quantum integral inequalities for convex functions","volume":"9","author":"Sudsudat","year":"2015","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_30","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":"Muhammad","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1186\/s13662-020-02559-3","article-title":"Quantum Hermite\u2013Hadamard inequality by means of a Green function","volume":"2020","author":"Muhammad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_32","unstructured":"Muhammad, A.N., Khalida, I.N., and Muhammad, U.A. (2015). Quantum analogues of Hermite\u2013Hadamard type inequalities for generalized convexity. Computation, Cryptography, and Network Security, Springer."},{"key":"ref_33","first-page":"13","article-title":"New estimates of-Ostrowski-type inequalities within a class of-polynomial prevexity of functions","volume":"2020","author":"Humaira","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1016\/j.amc.2014.11.090","article-title":"Some quantum estimates of Hermite\u2013Hadamard inequalities for convex functions","volume":"251","author":"Wenjun","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_35","first-page":"649","article-title":"Simpson type quantum integral inequalities for convex functions","volume":"9","author":"Tun","year":"2018","journal-title":"Miskolc Math. Notes"},{"key":"ref_36","first-page":"147","article-title":"Some \u03c6-analogues of Hermite\u2013Hadamard inequality for s-convex functions in the second sense and related estimates","volume":"48","author":"Latif","year":"2016","journal-title":"Punjab Univ. J. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"152","DOI":"10.3390\/math7020152","article-title":"Some quantum estimate of Hermite\u2013Hadamard inequalities for quasi-convex functions","volume":"7","author":"Hefeng","year":"2019","journal-title":"Mathematics"},{"key":"ref_38","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_39","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1186\/s13660-016-1024-1","article-title":"Integral inequalities via fractional quantum calculus","volume":"2016","author":"Sudsutad","year":"2016","journal-title":"J. Inequal. Appl."},{"key":"ref_40","first-page":"1","article-title":"Some integral inequalities via (p,q)-calculus on finite intervals","volume":"19","author":"Tun","year":"2016","journal-title":"RGMIA Res. Rep. Coll."},{"key":"ref_41","first-page":"1","article-title":"(p,q)-Integral inequalities","volume":"19","author":"Tun","year":"2016","journal-title":"RGMIA Res. Rep. Coll."},{"key":"ref_42","first-page":"969","article-title":"(p,q)-Hermite\u2013Hadamard inequalities and (p,q)-estimates for midpoint type inequalities via convex and quasi-convex functions","volume":"112","author":"Kunt","year":"2018","journal-title":"Rev. R. Acad. Cienc."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1186\/s13662-020-2512-7","article-title":"Fractional (p,q)-calculus","volume":"2020","author":"Soontharanon","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_44","unstructured":"Bukweli-Kyemba, J.D., and Hounkonnou, M.N. (2013). Quantum deformed algebra: Coherent states and special functions. arXiv."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Prabseang, J., Nonlaopon, K., and Tariboon, J. (2019). (p,q)-Hermite\u2013Hadamard inequalities for double integral and (p,q)-differentiable convex functions. Axioms, 8.","DOI":"10.3390\/axioms8020068"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Amer, M., Junjua, M.D., Hassain, S., and Shahxadi, G. (2019). (p,q)-estimates of Hermite\u2013Hadamard-type inequalities for coordinated convex and quasi convex function. Mathematics, 7.","DOI":"10.3390\/math7080683"},{"key":"ref_47","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_48","doi-asserted-by":"crossref","first-page":"384","DOI":"10.2991\/jnmp.1995.2.3-4.18","article-title":"Two-parameter deformation of the oscillator algebra and (p,q)-analog of two-dimensional conformal field theory","volume":"2","author":"Burban","year":"1995","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1080\/10652469408819035","article-title":"(p,q)-differentiation, (p,q)-integration, and (p,q)-hypergeometric functions related to quantum groups","volume":"2","author":"Burban","year":"1994","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"145","DOI":"10.55937\/sut\/1394548362","article-title":"R (p,q)-calculus: Differentiation and integration","volume":"49","author":"Hounkonnou","year":"2013","journal-title":"SUT J. Math."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"211","DOI":"10.2298\/PIM1716211A","article-title":"Applications of (p,q)-gamma function to Sz\u00e1sz durrmeyer operators","volume":"102","author":"Aral","year":"2017","journal-title":"Publ. L\u2019Inst. Math."},{"key":"ref_52","first-page":"268","article-title":"Representations of two parameter quantum algebras and (p,q)-special functions","volume":"335","author":"Sahai","year":"2007","journal-title":"Adv. Differ. Equ."},{"key":"ref_53","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_54","doi-asserted-by":"crossref","unstructured":"Usman, T., Saif, M., and Choi, J. (2020). Certain identities associated with (p,q)-binomial coefficients and (p,q)-Stirling polynomials of the second kind. Symmetry, 12.","DOI":"10.3390\/sym12091436"},{"key":"ref_55","unstructured":"Sadjang, P.N. (2015). On the (p,q)-gamma and the (p,q)-beta functions. arXiv."},{"key":"ref_56","first-page":"1562","article-title":"On two (p,q)-analogues of the laplace transform","volume":"23","author":"Sadjang","year":"2017","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"874","DOI":"10.1016\/j.amc.2015.04.090","article-title":"On (p,q)-analogues of Bernstein operators","volume":"266","author":"Mursaleen","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.amc.2016.02.008","article-title":"Erratum to \u201cOn (p,q)-analogues of Bernstein operators\u201d","volume":"278","author":"Mursaleen","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1186\/s13660-016-1111-3","article-title":"Erratum to \u201cSome approximation properties of (p,q)-Bernstein operators\u201d","volume":"2016","author":"Kang","year":"2016","journal-title":"J. Inequal. Appl."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1007\/s11785-016-0553-4","article-title":"Approximation by (p,q)-Lorentz polynomials on a compact disk","volume":"10","author":"Mursaleen","year":"2016","journal-title":"Complex Anal. Oper. Theory."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.amc.2015.12.006","article-title":"On (p,q)-analogues of Kantorovich type Bernstein-Stancu-Schurer operator","volume":"276","author":"Cai","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1016\/j.amc.2015.03.135","article-title":"Some approximation results of (p,q)-analogues of Bernstein-Stancu operators","volume":"264","author":"Mursaleen","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1186\/s13660-016-1045-9","article-title":"On Kantorovich modification of (p,q)-Baskakov operators","volume":"2016","author":"Acar","year":"2016","journal-title":"J. Inequal. Appl."},{"key":"ref_64","first-page":"25","article-title":"(p,q)-analogues of divided difference and Bernstein operators","volume":"2017","author":"Mursaleen","year":"2017","journal-title":"J. Nonlinear Funct. Anal."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/0097-3165(91)90020-H","article-title":"(p,q)-Stirling numbers and set partition statistics","volume":"56","author":"Wachs","year":"1991","journal-title":"J. Combin. Theory Ser. A."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1016\/0097-3165(94)90117-1","article-title":"\u03c3-restricted growth functions and (p,q)-Stirling numbers","volume":"68","author":"Wachs","year":"1994","journal-title":"J. Combin. Theory Ser."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"R84","DOI":"10.37236\/1837","article-title":"Rook theory, generalized Stirling numbers and (p,q)-analogues","volume":"11","author":"Remmel","year":"2004","journal-title":"Electron. J. Combin."},{"key":"ref_68","first-page":"474","article-title":"Generalized Stirling numbers, convolution formula and (p,q)-analogues","volume":"11","author":"Leroux","year":"1995","journal-title":"Can. J. Math."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s40590-016-0139-1","article-title":"(p,q)-beta functions and applications in approximation","volume":"24","author":"Gradimir","year":"2018","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Gasper, G., and Rahman, M. (2004). Basic Hypergeometric Series, Cambridge University Press.","DOI":"10.1017\/CBO9780511526251"},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Anastassiou, G.A. (2011). Intelligent Mathematics: Computational Analysis, Springer.","DOI":"10.1007\/978-3-642-17098-0"},{"key":"ref_72","doi-asserted-by":"crossref","unstructured":"Cerone, P., and Dragomir, S.S. (2010). Mathematical Inequalities, CRC Press.","DOI":"10.1201\/b10483"},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Pachpatte, B.G. (2012). Analytic Inequalities, Atlantis Press.","DOI":"10.2991\/978-94-91216-44-2"},{"key":"ref_74","unstructured":"Polya, G., and Szeq\u00f6, G. (1925). Aufaben und Lehrsatze aus der Analysis, Band 1. Die Grundlehren der Mathematischen Wissenschaften, Springer."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/504\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:38:11Z","timestamp":1760161091000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/3\/504"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,19]]},"references-count":74,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["sym13030504"],"URL":"https:\/\/doi.org\/10.3390\/sym13030504","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,19]]}}}