{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T07:42:34Z","timestamp":1769154154417,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T00:00:00Z","timestamp":1657843200000},"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>The celebrated Montgomery identity has been studied extensively since it was established. We found a novel version of the Montgomery identity when we were working inside the framework of p- and q-calculus. We acquire a Montgomery identity through a definite (p,q)-integral from these results. Consequently, we establish specific Ostrowski-type (p,q)-integral inequalities by using Montgomery identity. In addition to the well-known repercussions, this novel study provides an opportunity to set up new boundaries in the field of comparative literature. The research that is being proposed on the (p,q)-integral includes some fascinating results that demonstrate the superiority and applicability of the findings that have been achieved. This highly successful and valuable strategy is anticipated to create a new venue in the contemporary realm of special relativity and quantum theory. These mathematical inequalities and the approaches that are related to them have applications in the areas that deal with symmetry. Additionally, an application to special means is provided in the conclusion.<\/jats:p>","DOI":"10.3390\/sym14071449","type":"journal-article","created":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T01:57:11Z","timestamp":1657850231000},"page":"1449","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Montgomery Identity and Ostrowski-Type Inequalities for Generalized Quantum Calculus through Convexity and Their Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-3349","authenticated-orcid":false,"given":"Humaira","family":"Kalsoom","sequence":"first","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"additional","affiliation":[{"name":"Escuela de Ciencias F\u00edsicas y Matem\u00e1ticas, Facultad de Ciencias Naturales y Exactas, Pontificia Universidad Cat\u00f3lica del Ecuador, Sede Quito 17-01-2184, Ecuador"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8567-0676","authenticated-orcid":false,"given":"Muhammad Zainul","family":"Abidin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4242-9916","authenticated-orcid":false,"given":"Muhammad","family":"Marwan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"}]},{"given":"Zareen A.","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,15]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2002). Quantum Calculus, Universitext, Springer.","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_3","unstructured":"Lupas, A. (1987). A q-analogue of the Bernstein operator. Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca."},{"key":"ref_4","first-page":"511","article-title":"Bernstein polynomials based on the q-integers. The heritage of P.L. Chebyshev: A Festschrift in honor of the 70th-birthday of Professor T.J. Rivlin","volume":"4","author":"Phillips","year":"1997","journal-title":"Ann. Numer. 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