{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T10:26:11Z","timestamp":1771064771597,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T00:00:00Z","timestamp":1692144000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University","award":["RGP 2\/414\/44"],"award-info":[{"award-number":["RGP 2\/414\/44"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, we apply H\u00f6lder\u2019s inequality, Jensen\u2019s inequality, chain rule and the properties of convex functions and submultiplicative functions to develop an innovative category of dynamic Hardy-type inequalities on time scales delta calculus. A time scale, denoted by T, is any closed nonempty subset of R. In time scale calculus, results are unified and extended. As particular cases of our findings (when T=R), we have the continuous analogues of inequalities established in some the literature. Furthermore, we can find other inequalities in different time scales, such as T=N, which, to the best of the authors\u2019 knowledge, is a largely novel conclusion.<\/jats:p>","DOI":"10.3390\/axioms12080791","type":"journal-article","created":{"date-parts":[[2023,8,16]],"date-time":"2023-08-16T10:09:33Z","timestamp":1692180573000},"page":"791","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Novel Hardy-Type Inequalities with Submultiplicative Functions on Time Scales Using Delta Calculus"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6782-7908","authenticated-orcid":false,"given":"Haytham M.","family":"Rezk","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}]},{"given":"Ahmed I.","family":"Saied","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"given":"Maha","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, Sarat Abidah, King Khalid University, P.O. Box 64512, Abha 62529, Saudi Arabia"}]},{"given":"Belal A.","family":"Glalah","sequence":"additional","affiliation":[{"name":"Department of Basic Science, Higher Technological Institute, Tenth of Ramadan City 44629, Egypt"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,16]]},"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 of some points in the integral calculus, LX. An inequality between integrals","volume":"54","author":"Hardy","year":"1925","journal-title":"Mess. Math."},{"key":"ref_3","first-page":"285","article-title":"Generalization of inequalities of Hardy and Littlewood","volume":"31","author":"Leindler","year":"1970","journal-title":"Acta Sci. Math."},{"key":"ref_4","unstructured":"Opic, B., and Kufner, A. (1990). Hardy-Type Inequalities, Longman Scientific and Technical."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1016\/0022-247X(91)90316-R","article-title":"Weighted Hardy and Opial-type inequalities","volume":"160","author":"Sinnamon","year":"1991","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1007\/BF00970496","article-title":"Boundedness of linear integral operators on a class of monotone functions","volume":"32","author":"Stepanov","year":"1991","journal-title":"Siberian Math. J."},{"key":"ref_7","first-page":"141","article-title":"Elementary theorems concerning power series with positive coefficents and moment constants of positive functions","volume":"157","author":"Hardy","year":"1927","journal-title":"J. Math."},{"key":"ref_8","first-page":"12","article-title":"Notes of some points in the integral calculus, LXIV. Further inequalities between integrals","volume":"57","author":"Hardy","year":"1928","journal-title":"Mess. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1112\/jlms\/s1-3.3.205","article-title":"\u0170ber Reihen mit positiven Gliedern","volume":"3","author":"Knopp","year":"1928","journal-title":"J. Lond. Math. Soc."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.jat.2003.09.007","article-title":"On strenghtened Hardy and P\u00f3lya-Knopp\u2019s inequalities","volume":"125","author":"Persson","year":"2003","journal-title":"J. Approx. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.aml.2011.09.050","article-title":"Some Hardy type integral inequalities","volume":"25","author":"Sulaiman","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_13","first-page":"91","article-title":"Inequalities for averages of convex and superquadratic functions","volume":"5","author":"Abramovich","year":"2004","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"465","DOI":"10.4134\/JKMS.2013.50.3.465","article-title":"Time scales integral inequalities for superquadratic functions","volume":"50","author":"Baric","year":"2013","journal-title":"J. Korean Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2691","DOI":"10.1007\/s13398-019-00654-z","article-title":"More accurate dynamic Hardy-type inequalities obtained via superquadraticity","volume":"1","author":"Saker","year":"2019","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat."},{"key":"ref_16","first-page":"6427378","article-title":"Inequalities of hardy type via superquadratic functions with general kernels and measures for several variables on time scales","volume":"2020","author":"Rezk","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"61","DOI":"10.15352\/afa\/1396833503","article-title":"Time scales Hardy-type inequalities via superquadracity","volume":"5","author":"Oguntuase","year":"2014","journal-title":"Ann. Funct. Anal."},{"key":"ref_18","first-page":"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_19","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., O\u2019Regan, D., and Saker, S.H. (2016). Hardy Type Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-44299-0"},{"key":"ref_20","first-page":"9","article-title":"Hardy-type inequalities on time scales vie convexity in several variables","volume":"2013","author":"Donchev","year":"2013","journal-title":"ISRN Math. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"299","DOI":"10.7153\/jmi-07-28","article-title":"Minkowski and Beckenbach-Dresher inequalities and functionals on time scales","volume":"7","author":"Bibi","year":"2013","journal-title":"J. Math. Inequal."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Georgiev, S.G. (2016). Multiple Integration on Time Scales. Multivariable Dynamic Calculus on Time Scales, Springer.","DOI":"10.1007\/978-3-319-47620-9"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Kufner, A., and Persson, L.E. (2003). Weighted Inequalities of Hardy Type, World Scientific Publishing Co. Pte. Ltd.","DOI":"10.1142\/5129"},{"key":"ref_24","unstructured":"Kufner, A., Maligranda, L., and Persson, L.E. (2007). The Hardy Inequalities: About Its History and Some Related Results, Vydavatelsk\u00b4y Servis."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1155\/2022\/7997299","article-title":"On hardy-knopp type inequalities with kernels via time scale calculus","volume":"2022","author":"Rezk","year":"2022","journal-title":"J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1155\/2022\/7668860","article-title":"On a Refinement of Multidimensional Inequalities of Hardy-Type via Superquadratic and Subquadratic Functions","volume":"2022","author":"Zakarya","year":"2022","journal-title":"J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Saied, A.I., AlNemer, G., Zakarya, M., Cesarano, C., and Rezk, H.M. (2022). Some new generalized inequalities of Hardy type involving several functions on time scale nabla calculus. Axioms, 11.","DOI":"10.3390\/axioms11120662"},{"key":"ref_28","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_29","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., O\u2019Regan, D., and Saker, S.H. (2014). Dynamic Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-11002-8"},{"key":"ref_30","first-page":"87","article-title":"Convex functions on time scales","volume":"35","author":"Dinu","year":"2008","journal-title":"Ann. Univ. Craiova-Math. Comput. Sci. Ser."},{"key":"ref_31","unstructured":"S\u00e1ndor, J. (2011). Inequalities for multiplicative arithmetic functions. arXiv."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/791\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:34:41Z","timestamp":1760128481000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/8\/791"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,16]]},"references-count":31,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["axioms12080791"],"URL":"https:\/\/doi.org\/10.3390\/axioms12080791","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,16]]}}}