{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T09:41:18Z","timestamp":1769074878357,"version":"3.49.0"},"reference-count":17,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,22]],"date-time":"2022-11-22T00:00:00Z","timestamp":1669075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we establish several new generalized Hardy-type inequalities involving several functions on time-scale nabla calculus. Furthermore, we derive some new multidimensional Hardy-type inequalities on time scales nabla calculus. The main results are proved by applying Minkowski\u2019s inequality, Jensen\u2019s inequality and Arithmetic Mean\u2013Geometric Mean inequality. As a special case of our results, when T=R, we obtain refinements of some well-known continuous inequalities and when T=N, the results which are essentially new.<\/jats:p>","DOI":"10.3390\/axioms11120662","type":"journal-article","created":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T03:15:24Z","timestamp":1669173324000},"page":"662","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Some New Generalized Inequalities of Hardy Type Involving Several Functions on Time Scale Nabla Calculus"],"prefix":"10.3390","volume":"11","author":[{"given":"A. I.","family":"Saied","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"ALNemer","sequence":"additional","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4312-8330","authenticated-orcid":false,"given":"Mohammed","family":"Zakarya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6782-7908","authenticated-orcid":false,"given":"Haytham M.","family":"Rezk","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,22]]},"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. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1006\/jath.2002.3684","article-title":"On Carleman and Knopp\u2019s inequalities","volume":"117","author":"Kaijser","year":"2002","journal-title":"J. Approx. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.jat.2003.09.007","article-title":"On strengthened Hardy and P\u00f3lya-Knopp\u2019s inequalities","volume":"125","author":"Persson","year":"2003","journal-title":"J. Approx. Theory"},{"key":"ref_5","first-page":"403","article-title":"Hardy type inequalities via convexity","volume":"8","author":"Kaijser","year":"2005","journal-title":"Math. Inequal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2001). Dynamic Equations on Time Scales: An Introduction with Applications, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-0201-1"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2003). Advances in Dynamic Equations on Time Scales, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-8230-9"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2022\/7997299","article-title":"On Hardy\u2013Knopp type inequalities with kernels via time scale calculus","volume":"2022","author":"Rezk","year":"2022","journal-title":"J. Math."},{"key":"ref_9","first-page":"1","article-title":"Islam Abohela and Dumitru Baleanu, Refinement Multidimensional Dynamic Inequalities with General Kernels and Measures","volume":"306","author":"Saker","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Zakarya, M., Nemer, G.A.L., Saied, A.I., Butush, R., and Rezk, O.B.H.M. (2022). 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Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/662\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:24:21Z","timestamp":1760145861000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/662"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,22]]},"references-count":17,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120662"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120662","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,22]]}}}