{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T09:47:53Z","timestamp":1775382473003,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T00:00:00Z","timestamp":1684195200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University","award":["RSP2023R158"],"award-info":[{"award-number":["RSP2023R158"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Quantum calculus provides a significant generalization of classical concepts and overcomes the limitations of classical calculus in tackling non-differentiable functions. Implementing the q-concepts to obtain fresh variants of classical outcomes is a very intriguing aspect of research in mathematical analysis. The objective of this article is to establish novel Milne-type integral inequalities through the utilization of the Mercer inequality for q-differentiable convex mappings. In order to accomplish this task, we begin by demonstrating a new quantum identity of the Milne type linked to left and right q derivatives. This serves as a supporting result for our primary findings. Our approach involves using the q-equality, well-known inequalities, and convex mappings to obtain new error bounds of the Milne\u2013Mercer type. We also provide some special cases, numerical examples, and graphical analysis to evaluate the efficacy of our results. To the best of our knowledge, this is the first article to focus on quantum Milne\u2013Mercer-type inequalities and we hope that our methods and findings inspire readers to conduct further investigation into this problem.<\/jats:p>","DOI":"10.3390\/sym15051096","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T01:52:41Z","timestamp":1684288361000},"page":"1096","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Exploration of Quantum Milne\u2013Mercer-Type Inequalities with Applications"],"prefix":"10.3390","volume":"15","author":[{"given":"Bandar","family":"Bin-Mohsin","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 145111, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5212-6252","authenticated-orcid":false,"given":"Muhammad Zakria","family":"Javed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1019-9485","authenticated-orcid":false,"given":"Muhammad Uzair","family":"Awan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Awais Gul","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6105-2435","authenticated-orcid":false,"given":"Muhammad Aslam","family":"Noor","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 45550, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,16]]},"reference":[{"key":"ref_1","first-page":"73","article-title":"A variant of Jensen\u2019s inequality","volume":"4","author":"Mercer","year":"2003","journal-title":"J. Inequal Pure Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1186\/s13660-023-02921-5","article-title":"On new Milne-type inequalities for fractional integrals","volume":"2023","author":"Budak","year":"2023","journal-title":"J. Inequalities Appl."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Meftah, B., Lakhdari, A., Saleh, W., and Kili\u00e7man, A. (2023). Some New Fractal Milne-Type Integral Inequalities via Generalized Convexity with Applications. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020166"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ali, M.A., Zhang, Z., and Feckan, M. (2023). On Some Error Bounds for Milne\u2019s Formula in Fractional Calculus. Mathematics, 11.","DOI":"10.3390\/math11010146"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kac, V.G., and Cheung, P. (2002). Quantum Calculus, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"43","DOI":"10.3389\/fphy.2020.00043","article-title":"The falling body problem in quantum calculus","volume":"8","author":"Alanazi","year":"2020","journal-title":"Front. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Tariboon, J., and Ntouyas, S.K. (2013). Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ., 2013.","DOI":"10.1186\/1687-1847-2013-282"},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"742","DOI":"10.13001\/1081-3810.1684","article-title":"Refinements of the operator Jensen-Mercer inequality","volume":"26","author":"Kian","year":"2013","journal-title":"Electron. J. Linear Algebra"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2425","DOI":"10.2298\/FIL2107425O","article-title":"Hermite-Hadamard-Mercer type inequalities for fractional integrals","volume":"35","author":"Ogulmus","year":"2021","journal-title":"Filomat"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5193","DOI":"10.3934\/math.2020334","article-title":"Hermite-Jensen-Mercer type inequalities via \u03a8-Riemann-Liouville k -fractional integrals","volume":"5","author":"Butt","year":"2020","journal-title":"AIMS Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"9397","DOI":"10.3934\/math.2021546","article-title":"Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions","volume":"6","author":"Ali","year":"2021","journal-title":"AIMS Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3203","DOI":"10.3934\/math.2022317","article-title":"Some new generalized k-fractional Hermite-Hadamard-Mercer type integral inequalities and their applications","volume":"7","author":"Awan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_14","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_15","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_16","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_17","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., and Vivas-Cortez, M. (2022). (q1,q2)-Ostrowski-Type Integral Inequalities Involving Property of Generalized Higher-Order Strongly n-Polynomial Preinvexity. Symmetry, 14.","DOI":"10.3390\/sym14040717"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1515\/ms-2023-0029","article-title":"A new version of q-Hermite-Hadamard\u2019s midpoint and trapezoid type inequalities for convex functions","volume":"73","author":"Ali","year":"2023","journal-title":"J. Math. Slovaca"},{"key":"ref_19","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_20","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Sarikaya, M.Z., and Baleanu, D. (2020). On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals. Symmetry, 12.","DOI":"10.3390\/sym12040595"},{"key":"ref_21","first-page":"1","article-title":"Quantum Hermite-Hadamard inequality by means of A Green function","volume":"2020","author":"Khan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_22","unstructured":"Saleh, W., Meftah, B., and Lakhdari, A. (2023). Quantum dual Simpson type inequalities for q-differentiable convex functions. Int. J. Nonlinear Anal. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Erden, S., Iftikhar, S., Kumam, P., and Thounthong, P. (2020). On error estimations of Simpson\u2019s second type quadrature formula. Math. Methods Appl. Sci., 1\u201313.","DOI":"10.1002\/mma.7019"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Raees, M., and Anwar, M. (2023). New Estimation of Error in the Hadamard Inequality Pertaining to Coordinated Convex Functions in Quantum Calculus. Symmetry, 15.","DOI":"10.3390\/sym15020301"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.11121\/ijocta.2023.1258","article-title":"Certain Saigo type fractional integral inequalities and their q-analogues","volume":"13","author":"Jain","year":"2023","journal-title":"Int. J. Optim. Control Theor. Appl."},{"key":"ref_26","first-page":"122","article-title":"Fractional quantum Hermite-Hadamard type inequalities","volume":"8","author":"Kunt","year":"2020","journal-title":"Konuralp J. Math."},{"key":"ref_27","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."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1186\/s13662-020-03163-1","article-title":"Quantum Hermite\u2013Hadamard-type inequalities for functions with convex absolute values of second q\u03c52-derivatives","volume":"2021","author":"Ali","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2941","DOI":"10.1002\/mma.8680","article-title":"Some Hermite-Hadamard\u2019s type local fractional integral inequalities for generalized \u03b3-preinvex function with applications","volume":"46","author":"Bibi","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Chasreechai, S., Ali, M.A., Ashraf, M.A., Sitthiwirattham, T., Etemad, S., Sen, M.D.L., and Rezapour, S. (2023). On New Estimates of q-Hermite-Hadamard Inequalities with Applications in Quantum Calculus. Axioms, 12.","DOI":"10.3390\/axioms12010049"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Teklu, B., Olivares, S., and Paris, M.G. (2009). Bayesian estimation of one-parameter qubit gates. J. Phys. At. B Mol. Opt. Phys., 42.","DOI":"10.1088\/0953-4075\/42\/3\/035502"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Brivio, D., Cialdi, S., Vezzoli, S., Gebrehiwot, B.T., Genoni, M.G., Olivares, S., and Paris, M.G. (2010). Experimental estimation of one-parameter qubit gates in the presence of phase diffusion. Phys. Rev. A, 81.","DOI":"10.1103\/PhysRevA.81.012305"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Xu, K., and Heo, J. (2010). New functional glasses containing semiconductor quantum dots. Phys. Scr., 2010.","DOI":"10.1088\/0031-8949\/2010\/T139\/014062"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1096\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:36:17Z","timestamp":1760124977000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1096"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,16]]},"references-count":33,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15051096"],"URL":"https:\/\/doi.org\/10.3390\/sym15051096","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,16]]}}}