{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T03:58:38Z","timestamp":1770523118790,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,27]],"date-time":"2023-08-27T00:00:00Z","timestamp":1693094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Researchers Supporting","award":["RSPD2023R1004"],"award-info":[{"award-number":["RSPD2023R1004"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this research, we aim to explore generalizations of Hardy-type inequalities using nabla H\u00f6lder\u2019s inequality, nabla Jensen\u2019s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature.<\/jats:p>","DOI":"10.3390\/sym15091656","type":"journal-article","created":{"date-parts":[[2023,8,28]],"date-time":"2023-08-28T02:23:25Z","timestamp":1693189405000},"page":"1656","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales"],"prefix":"10.3390","volume":"15","author":[{"given":"Haytham M.","family":"Rezk","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}]},{"given":"Mahmoud I.","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8870-9692","authenticated-orcid":false,"given":"Oluwafemi Samson","family":"Balogun","sequence":"additional","affiliation":[{"name":"Department of Computing, University of Eastern Finland, 70211 Kuopio, Finland"}]},{"given":"Ahmed I.","family":"Saied","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1007\/BF01199965","article-title":"Notes on a theorem of Hilbert","volume":"6","author":"Hardy","year":"1920","journal-title":"Math. Z."},{"key":"ref_2","first-page":"150","article-title":"Notes on some points in the integral calculus, LX. An inequality between integrals","volume":"54","author":"Hardy","year":"1925","journal-title":"Mess. 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