{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:38:20Z","timestamp":1760233100837,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,17]],"date-time":"2022-12-17T00:00:00Z","timestamp":1671235200000},"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 established some new \u03b1-conformable dynamic inequalities of Hardy\u2013Knopp type. Some new generalizations of dynamic inequalities of \u03b1-conformable Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy\u2019s inequality for several functions of \u03b1-conformable calculus. Our results are proved by using two-dimensional dynamic Jensen\u2019s inequality and Fubini\u2019s theorem on time scales. When \u03b1=1, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we derived Hardy\u2019s inequality for T=R,T=Z and T=hZ. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.<\/jats:p>","DOI":"10.3390\/sym14122674","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T06:58:29Z","timestamp":1671433109000},"page":"2674","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamic Inequalities of Two-Dimensional Hardy Type via Alpha-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, Cairo 11884, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8846-0487","authenticated-orcid":false,"given":"Alaa A.","family":"El-Bary","sequence":"additional","affiliation":[{"name":"Basic and Applied Science Institute, Arab Academy for Science, Technology and Maritime Transport, Alexandria P.O. Box 1029, Egypt"},{"name":"National Committee for Mathematics, Academy of Scientific Research and Technology, Cairo 11516, Egypt"},{"name":"Council of Future Studies and Risk Management, Academy of Scientific Research and Technology, Cairo 11516, 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\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,17]]},"reference":[{"key":"ref_1","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_2","first-page":"150","article-title":"Notes on some points in the integral calculus (lx)","volume":"54","author":"Hardy","year":"1925","journal-title":"Messenger Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1515\/crll.1927.157.141","article-title":"Elementary theorems concerning power series with positive coeficients and moment constants of positive functions","volume":"157","author":"Littlewood","year":"1927","journal-title":"J. Reine Angew. Math."},{"key":"ref_4","unstructured":"Hardy, G.H., Littlewood, J.E., and Polya, G. (1952). Inequalities, Cambridge University Press. [2nd ed.]."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"834","DOI":"10.1137\/0514064","article-title":"Weighted norm inequalities for certain integral operators","volume":"14","author":"Andersen","year":"1983","journal-title":"SIAM J. Math. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"9","DOI":"10.4064\/sm-72-1-9-26","article-title":"Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions","volume":"72","author":"Andersen","year":"1982","journal-title":"Stud. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1093\/qmath\/38.4.401","article-title":"Some elementary inequalities","volume":"38","author":"Bennett","year":"1987","journal-title":"Quart. J. Math. Oxf. Ser."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Georgiev, S.G. (2020). Integral Inequalities on Time Scales, De Gruyter.","DOI":"10.1515\/9783110705553"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1002\/mma.6805","article-title":"On the number of eigenvalues for parameter-dependent diffusion problem on time scales","volume":"44","author":"Gulsen","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kufner, A., and Persson, L.-E. (2003). Weighted Inequalities of Hardy Type, World Scientific Publishing Co., Inc.","DOI":"10.1142\/5129"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/BF03323153","article-title":"Analysis on measure chainsa unified approach to continuous and discrete calculus","volume":"18","author":"Hilger","year":"1990","journal-title":"Results Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2001). Dynamic Equations on Time Scales, Birkhauser Boston, Inc.. An Introduction with Applications.","DOI":"10.1007\/978-1-4612-0201-1"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2003). Advances in Dynamic Equations on Time Scales, Birkhauser Boston, Inc.","DOI":"10.1007\/978-0-8176-8230-9"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Rehak, P. (2005). Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl., 495\u2013507.","DOI":"10.1155\/JIA.2005.495"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1002\/mana.201300010","article-title":"Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales","volume":"287","author":"Saker","year":"2014","journal-title":"Math. Nachr."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1017\/S0004972717000478","article-title":"Some reverse dynamic inequalities on time scales","volume":"96","author":"Agarwal","year":"2017","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1186\/s13662-020-02857-w","article-title":"Some reverse inequalities of Hardy type on time scales","volume":"2020","author":"Khan","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1186\/s13660-020-02497-4","article-title":"On structure of discrete muchenhoupt and discrete gehring classes","volume":"2020","author":"Saker","year":"2020","journal-title":"J. Inequalities Appl."},{"key":"ref_20","first-page":"477","article-title":"Hardy-Knopp-type inequalities on time scales","volume":"17","author":"Ozkan","year":"2008","journal-title":"Dynam. Syst. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.jksus.2015.05.003","article-title":"A conformable fractional calculus on arbitrary time scales","volume":"28","author":"Benkhettou","year":"2016","journal-title":"J. King Saud-Univ.-Sci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sakerr, S.H., Kenawy, M., AlNemer, G.H., and Zakarya, M. (2020). Some fractional dynamic inequalities of hardy\u2019s type via conformable calculus. Mathematics, 8.","DOI":"10.3390\/math8030434"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zakaryaed, M., Altanji, M., AlNemer, G.H., El-Hamid, A., Hoda, A., Cesarano, C., and Rezk, H.M. (2021). Fractional reverse coposn\u2019s inequalities via conformable calculus on time scales. Symmetry, 13.","DOI":"10.3390\/sym13040542"},{"key":"ref_24","first-page":"579","article-title":"Multiple integration on time scales","volume":"14","author":"Bohner","year":"2005","journal-title":"Dynam. Syst. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Agarwal, R., O\u2019Regan, D., and Saker, S. (2014). Dynamic Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-11002-8"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2674\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:12Z","timestamp":1760146992000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2674"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,17]]},"references-count":25,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["sym14122674"],"URL":"https:\/\/doi.org\/10.3390\/sym14122674","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,17]]}}}