{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T10:48:11Z","timestamp":1769078891045,"version":"3.49.0"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T00:00:00Z","timestamp":1662336000000},"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>We prove new Hardy\u2013Copson-type (\u03b3,a)-nabla fractional dynamic inequalities on time scales. Our results are proven by using Keller\u2019s chain rule, the integration by parts formula, and the dynamic H\u00f6lder inequality on time scales. When \u03b3=1, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we obtain new continuous and discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.<\/jats:p>","DOI":"10.3390\/sym14091847","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T09:51:09Z","timestamp":1662630669000},"page":"1847","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Dynamic Hardy\u2013Copson-Type Inequalities via (\u03b3,a)-Nabla-Conformable Derivatives on Time Scales"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2822-4092","authenticated-orcid":false,"given":"Ahmed A.","family":"El-Deeb","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt"}]},{"given":"Samer D.","family":"Makharesh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0387-921X","authenticated-orcid":false,"given":"Jan","family":"Awrejcewicz","sequence":"additional","affiliation":[{"name":"Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1\/15 Stefanowski St., 90-924 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0634-2370","authenticated-orcid":false,"given":"Ravi P.","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363-8202, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,5]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. (1952). Inequalities, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"942973","DOI":"10.1155\/JIA.2005.495","article-title":"Hardy inequality on time scales and its application to half-linear dynamic equations","volume":"2005","year":"2005","journal-title":"J. Inequal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s40840-015-0300-4","article-title":"Hardy and Littlewood inequalities on time scales","volume":"39","author":"Saker","year":"2016","journal-title":"Bull. Malaysian Math. Sci. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.21608\/JOMES.2018.9457","article-title":"Some Gronwall-bellman type inequalities on time scales for Volterra-Fredholm dynamic integral equations","volume":"26","year":"2018","journal-title":"J. Egypt. Math. 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Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1007\/BF01199965","article-title":"Note on a theorem of Hilbert","volume":"6","author":"Hardy","year":"1920","journal-title":"Math. Z."},{"key":"ref_10","first-page":"12","article-title":"Notes on some points in the integral calculus (LXIT)","volume":"57","author":"Hardy","year":"1928","journal-title":"Messenger Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1112\/jlms\/s1-3.1.49","article-title":"Note on series of positive terms","volume":"1","author":"Copson","year":"1928","journal-title":"J. Lond. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1017\/S0308210500017868","article-title":"Some Integral Inequalities","volume":"75","author":"Copson","year":"1976","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2001). Dynamic Equations on Time Scales: An Introduction with Applications, Springer Science and Business Media.","DOI":"10.1007\/978-1-4612-0201-1"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A.C. (2002). Advances in Dynamic Equations on Time Scales, Springer Science and Business Media.","DOI":"10.1007\/978-0-8176-8230-9"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1040","DOI":"10.3906\/mat-2011-38","article-title":"Hardy-Copson type inequalities for nabla time scale calculus","volume":"45","author":"Kayar","year":"2021","journal-title":"Turk. J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1186\/s13662-021-03385-x","article-title":"A new conformable nabla derivative and its application on arbitrary time scales","volume":"2021","author":"Noorani","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.cam.2014.10.016","article-title":"On conformable fractional calculus","volume":"279","author":"Abdeljawad","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"El-Deeb, A.A., and Awrejcewicz, J. (2021). Novel Fractional Dynamic Hardy\u2013Hilbert-Type Inequalities on Time Scales with Applications. 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