{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T21:01:04Z","timestamp":1780606864656,"version":"3.54.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T00:00:00Z","timestamp":1763337600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T00:00:00Z","timestamp":1763337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s12190-025-02672-0","type":"journal-article","created":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T14:20:54Z","timestamp":1763389254000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A quantum calculus approach to Boole\u2019s formula type inequalities with computational analysis and applications"],"prefix":"10.1007","volume":"72","author":[{"given":"Mubashir","family":"Hussain","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdul","family":"Mateen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H\u00fcseyin","family":"Budak","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Asna","family":"Aslam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9868-7079","authenticated-orcid":false,"given":"Haibo","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,11,17]]},"reference":[{"key":"2672_CR1","first-page":"193","volume":"41","author":"F.H. Jackson","year":"1910","unstructured":"Jackson, F.H.: On $$q$$-definite integrals. Q. J. Pure Appl. Math. 41, 193\u2013203 (1910)","journal-title":"Q. J. Pure Appl. Math."},{"key":"2672_CR2","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1017\/S0080456800002751","volume":"46","author":"F.H. Jackson","year":"1908","unstructured":"Jackson, F.H.: On $$q$$-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253\u2013281 (1908)","journal-title":"Trans. R. Soc. Edinb."},{"key":"2672_CR3","doi-asserted-by":"publisher","first-page":"020001","DOI":"10.1063\/1.5097511","volume":"2095","author":"O.P. Ahuja","year":"2019","unstructured":"Ahuja, O.P., \u015eetinkaya, A.: Use of quantum calculus approach in mathematical sciences and its role in geometric function theory. AIP Conf. Proc. 2095, 020001 (2019)","journal-title":"AIP Conf. Proc."},{"key":"2672_CR4","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s40995-019-00815-0","volume":"44","author":"H.M. Srivastava","year":"2020","unstructured":"Srivastava, H.M.: Operators of basic (or $$q$$-) Calculus and fractional $$q$$-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. Sci. 44, 327\u2013344 (2020)","journal-title":"Iran. J. Sci. Technol. Trans. Sci."},{"key":"2672_CR5","doi-asserted-by":"publisher","first-page":"669","DOI":"10.4236\/am.2021.128047","volume":"12","author":"M.O. Sadik","year":"2021","unstructured":"Sadik, M.O., Orie, B.O.: Application of $$q$$-calculus to the solution of partial $$q$$-differential equations. Appl. Math. 12, 669\u2013678 (2021)","journal-title":"Appl. Math."},{"key":"2672_CR6","volume-title":"The History of Q-Calculus and New Method","author":"T. Ernst","year":"2000","unstructured":"Ernst, T.: The History of Q-Calculus and New Method. Department of Mathematics, Uppsala University. (2000)"},{"key":"2672_CR7","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1088\/0954-3899\/16\/8\/006","volume":"16","author":"P.P. Raychev","year":"1990","unstructured":"Raychev, P.P., Roussev, R.P., Smirnov, Y.F.: The quantum algebra $$SU_{q}(2)$$ and rotational spectra of deformed nuclei. J. Phys. G: Nucl. Part. Phys. 16, 137\u2013141 (1990)","journal-title":"J. Phys. G: Nucl. Part. Phys."},{"key":"2672_CR8","first-page":"1","volume":"140","author":"B. Ahmad","year":"2012","unstructured":"Ahmad, B., Ntouyas, S.K., Purnaras, I.K.: Existence results for nonlocal boundary value problems of nonlinear fractional $$q$$-difference equations. Adv. Differ. Equ. 140, 1\u201315 (2012)","journal-title":"Adv. Differ. Equ."},{"key":"2672_CR9","volume-title":"New York, Q-Hypergeometric Functions and Applications","author":"H. Exton","year":"1983","unstructured":"Exton, H.: New York, Q-Hypergeometric Functions and Applications. Halstead Press. (1983)"},{"key":"2672_CR10","doi-asserted-by":"publisher","first-page":"1733","DOI":"10.1090\/S0002-9939-09-10185-5","volume":"138","author":"M. El-Shahed","year":"2010","unstructured":"El-Shahed, M., Hassan, H.A.: Positive solutions of $$q$$-difference equation. Proc. Amer. Math. Soc. 138, 1733\u20131738 (2010)","journal-title":"Proc. Amer. Math. Soc."},{"key":"2672_CR11","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/j.cam.2005.06.009","volume":"193","author":"A. Dobrogowska","year":"2006","unstructured":"Dobrogowska, A., Odzijewicz, A.: Second order $$q$$-difference equation solvable by factorization method. J. Comput. Appl. Math. 193, 319\u2013346 (2006)","journal-title":"J. Comput. Appl. Math."},{"key":"2672_CR12","doi-asserted-by":"crossref","unstructured":"Bermudo, S., K\u00f3rus, P., N\u00e1poles Vald\u00e9s, J.E.: On $$q$$-Hermite\u2013Hadamard inequalities for general convex functions. Acta Math. Hungar. 162, 364\u2013374","DOI":"10.1007\/s10474-020-01025-6"},{"key":"2672_CR13","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1515\/ms-2023-0029","volume":"73","author":"M.A. Ali","year":"2023","unstructured":"Ali, M.A., Budak, H., Feckan, M., Khan, S.: A new version of $$q$$-Hermite\u2013Hadamard\u2019s midpoint and trapezoid type inequalities for convex functions. Math. Slovaca 73, 369\u2013386 (2023)","journal-title":"Math. Slovaca"},{"key":"2672_CR14","doi-asserted-by":"publisher","first-page":"1845","DOI":"10.1002\/mma.7889","volume":"45","author":"M.A. Ali","year":"2022","unstructured":"Ali, M.A., Budak, H., Zhang, Z.: A new extension of quantum Simpson\u2019s and quantum Newton\u2019s type inequalities for quantum differentiable convex functions. Math. Methods Appl. Sci. 45, 1845\u20131863 (2022)","journal-title":"Math. Methods Appl. Sci."},{"key":"2672_CR15","doi-asserted-by":"publisher","first-page":"199","DOI":"10.22199\/issn.0717-6279-2021-01-0013","volume":"40","author":"H. Budak","year":"2021","unstructured":"Budak, H.: Some trapezoid and midpoint type inequalities for newly defined quantum integrals. Proyecciones 40, 199\u2013215 (2021)","journal-title":"Proyecciones"},{"key":"2672_CR16","unstructured":"Dwilewicz, R.J.: A short history of convexity. Differ. Geom.-Dyn. Syst. (2009)"},{"key":"2672_CR17","volume-title":"Numerical Analysis and Scientific Computation","author":"J.J. Leader","year":"2004","unstructured":"Leader, J.J.: Numerical Analysis and Scientific Computation. Addison Wesley, (2004)"},{"key":"2672_CR18","volume-title":"An Introduction to the History of Mathematics","author":"H. Eves","year":"1990","unstructured":"Eves, H.: An Introduction to the History of Mathematics. Saunders, (1990)"},{"key":"2672_CR19","doi-asserted-by":"publisher","first-page":"541","DOI":"10.36045\/j.bbms.240612","volume":"31","author":"A. Mateen","year":"2024","unstructured":"Mateen, A., Zhang, Z., Toseef, M., Ali, M.A.: A new version of Boole\u2019s formula type inequalities in multiplicative calculus with application to quadrature formula. Bull. Belg. Math. Soc. Simon Stevin 31, 541\u2013562 (2024)","journal-title":"Bull. Belg. Math. Soc. Simon Stevin"},{"key":"2672_CR20","doi-asserted-by":"publisher","first-page":"551","DOI":"10.3390\/math13040551","volume":"13","author":"W. Haider","year":"2025","unstructured":"Haider, W., Mateen, A., Budak, H., Shehzadi, A., Ciurdariu, L.: Novel fractional Boole\u2019s-type integral inequalities via Caputo fractional operator and their implications in numerical analysis. Mathematics 13, 551 (2025)","journal-title":"Mathematics"},{"key":"2672_CR21","doi-asserted-by":"publisher","first-page":"2439","DOI":"10.2298\/FIL2507439T","volume":"39","author":"M.T. Khan","year":"2025","unstructured":"Khan, M.T., Toseef, M., Kashuri, A.: Quantum Weddle\u2019s type inequalities for convex functions with their computational analysis and applications. Filomat 39, 2439\u20132455 (2025)","journal-title":"Filomat"},{"key":"2672_CR22","doi-asserted-by":"publisher","first-page":"115973","DOI":"10.1016\/j.chaos.2024.115973","volume":"192","author":"A. Mateen","year":"2025","unstructured":"Mateen, A., Zhang, Z., Budak, H., \u00d6zcan, S.: Some novel inequalities of Weddle\u2019s formula type for Riemann\u2013Liouville fractional integrals with their applications to numerical integration. Chaos Solitons Fract. 192, 115973 (2025)","journal-title":"Chaos Solitons Fract."},{"key":"2672_CR23","unstructured":"Hermite, C.: Sur deux limites d\u2019une integrale de finie. Mathesis 82, (1883)"},{"key":"2672_CR24","first-page":"171","volume":"58","author":"J. Hadamard","year":"1893","unstructured":"Hadamard, J.: Etude sur les fonctions entiees et en particulier d\u2019une fonction consideree par Riemann. J. Math. Pures. Appl. 58, 171\u2013215 (1893)","journal-title":"J Math Pures Appl."},{"key":"2672_CR25","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/S0893-9659(98)00086-X","volume":"11","author":"S.S. Dragomir","year":"1998","unstructured":"Dragomir, S.S., Agarwal, P.R.: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoid formula. Appl. Math. Lett. 11, 91\u201395 (1998)","journal-title":"Appl. Math. Lett."},{"key":"2672_CR26","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/S0096-3003(02)00657-4","volume":"147","author":"U.S. Kirmaci","year":"2004","unstructured":"Kirmaci, U.S.: Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147, 137\u2013146 (2004)","journal-title":"Appl. Math. Comput."},{"key":"2672_CR27","doi-asserted-by":"crossref","unstructured":"Alp, N., Sarikaya, M.Z., Kunt, M., \u0130scan, \u0130.: $$q$$-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. J. King Saud. Univ. Sci. 30, 193\u2013203 (2018)","DOI":"10.1016\/j.jksus.2016.09.007"},{"key":"2672_CR28","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1007\/s10474-020-01025-6","volume":"162","author":"S. Bermudo","year":"2020","unstructured":"Bermudo, S., Korus, P., N\u00e1poles Vald\u00e9s, J.E.: On $$q$$-Hermite-Hadamard inequalities for general convex functions. Acta Math. Hungar. 162, 364\u2013374 (2020)","journal-title":"Acta Math. Hungar."},{"key":"2672_CR29","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1016\/j.amc.2014.11.090","volume":"251","author":"M.A. Noor","year":"2015","unstructured":"Noor, M.A., Noor, K.I., Awan, M.U.: Some quantum estimates for Hermite-Hadamard inequalities. Appl. Math. Comput. 251, 675\u2013679 (2015)","journal-title":"Appl. Math. Comput."},{"key":"2672_CR30","doi-asserted-by":"publisher","first-page":"199","DOI":"10.22199\/issn.0717-6279-2021-01-0013","volume":"40","author":"H. Budak","year":"2021","unstructured":"Budak, H.: Some trapezoid and midpoint type inequalities for newly defined quantum integrals. Proyecciones J. Math. 40, 199\u2013215 (2021)","journal-title":"Proyecciones J. Math."},{"key":"2672_CR31","doi-asserted-by":"publisher","first-page":"49","DOI":"10.3390\/axioms12010049","volume":"12","author":"S. Chasreechai","year":"2023","unstructured":"Chasreechai, S., Ali, M.A., Ashraf, M.A., Sitthiwirattham, T., Etemad, S., Sen, M.D.L., Rezapour, S.: On new estimates of $$q$$-Hermite\u2013Hadamard inequalities with applications in quantum calculus. Axioms 12, 49 (2023)","journal-title":"Axioms"},{"key":"2672_CR32","doi-asserted-by":"publisher","first-page":"508","DOI":"10.1016\/j.cjph.2025.03.013","volume":"95","author":"H. Kalsoom","year":"2025","unstructured":"Kalsoom, H., Almohsen, B.: Quantum calculus approaches to Owstrowski-type inequalities for strongly $$n$$-polymial convex functions with applications to quantum physics. Chin. J. Phys. 95, 508\u2013528 (2025)","journal-title":"Chin. J. Phys."},{"key":"2672_CR33","doi-asserted-by":"publisher","first-page":"3329","DOI":"10.2298\/FIL2410329M","volume":"38","author":"A. Mateen","year":"2024","unstructured":"Mateen, A., Zhang, Z., Ali, M.A.: Some Milne\u2019s rule type inequalities for convex functions with their computational analysis on quantum calculus. Filomat 38, 3329\u20133345 (2024)","journal-title":"Filomat"},{"key":"2672_CR34","first-page":"736","volume":"14","author":"M.A. Ali","year":"2021","unstructured":"Ali, M.A., Budak, H., Nanlaopon, K., Abdullah, Z.: Simpson\u2019s and Newton\u2019s inequalities for $$(a,m)$$-convex functions via quantum calculus. Symmetry 14, 736 (2021)","journal-title":"Symmetry"},{"key":"2672_CR35","doi-asserted-by":"crossref","unstructured":"Sial, I.B., Mei, S., Ali, M.A., Nanlaopon, K.: On some generalized Simpson\u2019s and Newton\u2019s inequalities for $$(a,m)$$-convex functions in $$q$$-calculus. Mathematics 3266 (2021)","DOI":"10.3390\/math9243266"},{"key":"2672_CR36","doi-asserted-by":"crossref","unstructured":"Mohammadi, H., Baleanu, D., Etemad, S., Rezapour, S.: Criteria for existence of solutions for a Liouville\u2013Caputo boundary value problem via generalized Gronwall\u2019s inequality. J. Inequal. Appl. 1\u201319 (2021)","DOI":"10.1186\/s13660-021-02562-6"},{"key":"2672_CR37","doi-asserted-by":"publisher","first-page":"479","DOI":"10.3390\/sym15020479","volume":"15","author":"M. Mukhtar","year":"2023","unstructured":"Mukhtar, M., Yaqoob, M., Samraiz, M., Shabbir, I., Etemad, S., Sen, M.D.L., Rezapour, S.: Novel mean-type inequalities via generalized Riemann-type fractional integral for composite convex functions: some special examples. Symmetry 15, 479 (2023)","journal-title":"Symmetry"},{"key":"2672_CR38","doi-asserted-by":"publisher","first-page":"2682","DOI":"10.3390\/sym14122682","volume":"14","author":"M. Samraiz","year":"2022","unstructured":"Samraiz, M., Malik, M., Saeed, K., Naheed, S., Etemad, S., Sen, M.D.L., Rezapour, S.: A study on the modified form of Riemann-type fractional inequalities via convex functions and related applications. Symmetry 14, 2682 (2022)","journal-title":"Symmetry"},{"key":"2672_CR39","doi-asserted-by":"crossref","unstructured":"Tariboon, J., Ntouyas, S.K.: Quantum calculus on finite intervals and applications to impulsive difference alignments. Adv. Differ. Equ. 282 (2013)","DOI":"10.1186\/1687-1847-2013-282"},{"key":"2672_CR40","volume-title":"Fractional Differential Equations","author":"I. Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional Differential Equations. Academic (1999)"},{"key":"2672_CR41","volume-title":"Fractional Calculus: integral and Differential Equations of Fractional Order","author":"R. Gorenflo","year":"1997","unstructured":"Gorenflo, R., Mainardi, F.: Fractional Calculus: integral and Differential Equations of Fractional Order. Springer, (1997)"},{"key":"2672_CR42","unstructured":"Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198 (1998)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02672-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02672-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02672-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T16:47:15Z","timestamp":1767890835000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02672-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,17]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["2672"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02672-0","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,17]]},"assertion":[{"value":"10 June 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 June 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 October 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors confirm that there are no conflicts of interest associated with this research.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"14"}}