{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T05:25:52Z","timestamp":1778131552146,"version":"3.51.4"},"reference-count":29,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T00:00:00Z","timestamp":1598313600000},"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>Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate some integral and discrete inequalities that have been demonstrated in the literature and also extend some of the dynamic inequalities that have been achieved in time scales.<\/jats:p>","DOI":"10.3390\/sym12091410","type":"journal-article","created":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T09:30:07Z","timestamp":1598347807000},"page":"1410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Some Dynamic Hilbert-Type Inequalities on Time Scales"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"AlNemer","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia"}]},{"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":"Hoda A.","family":"Abd El-Hamid","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7556-8942","authenticated-orcid":false,"given":"Praveen","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Anand International College of Engineering, Jaipur 303012, Rajasthan, India"}]},{"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":[[2020,8,25]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. (1934). Inequalities, Cambridge University Press. [2nd ed.]."},{"key":"ref_2","first-page":"5417084","article-title":"Dynamic Inequalities in Quotients with General Kernels and Measures","volume":"2020","author":"Rezk","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Saker, S.H., Kenawy, M.R., AlNemer, G., and Zakarya, M. (2020). Some Fractional Dynamic Inequalities of Hardys Type Via Conformable Calculus. Mathematics, 8.","DOI":"10.3390\/math8030434"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1006\/jmaa.2000.7006","article-title":"Some Hardy-type inequalities","volume":"250","author":"Cheung","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1090\/S0002-9939-98-04444-X","article-title":"On the extended Hilbert\u2019s inequality","volume":"126","author":"Gao","year":"1998","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1006\/jmaa.1999.6373","article-title":"On New Extensions of Hilbert\u2019s Integral Inequality","volume":"235","author":"Jichang","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1006\/jmaa.2000.7350","article-title":"New Hilbert-Pachpatte type integral inequalities","volume":"257","author":"Handley","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1006\/jmaa.1998.6043","article-title":"On some new inequalities similar to Hilbert\u2019s inequality","volume":"226","author":"Pachpatte","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1006\/jmaa.2000.6860","article-title":"On new generalizations of Hilbert\u2019s inequality","volume":"248","author":"Yang","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"311","DOI":"10.5556\/j.tkjm.31.2000.389","article-title":"A Hilbert type inequality","volume":"31","author":"Handley","year":"2000","journal-title":"Tamkang J. Math."},{"key":"ref_11","first-page":"625","article-title":"On the way of weight coefficient and research for the Hilbert-type inequalities","volume":"6","author":"Yang","year":"2003","journal-title":"Math. Appl."},{"key":"ref_12","first-page":"413","article-title":"Generalization on two new Hilbert type inequalities","volume":"20","author":"Zhao","year":"2000","journal-title":"J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"293","DOI":"10.5556\/j.tkjm.29.1998.4258","article-title":"A note on Hilbert type inequality","volume":"29","author":"Pachpatte","year":"1998","journal-title":"Tamkang J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1006\/jmaa.1999.6646","article-title":"Inequalities Similar to Certain Extensions of Hilbert\u2019s Inequality","volume":"243","author":"Pachpatte","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"377","DOI":"10.4134\/BKMS.2002.39.3.377","article-title":"An Analogue of Hilbert\u2019s inequality and its extensions","volume":"39","author":"Kim","year":"2002","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_16","unstructured":"Hilger, S. (1988). Ein Ma\u00dfkettenkalk\u00fcl mit Anwendung auf Zentrumsmannigfaltigkeiten. [Ph.D. Thesis, Universit\u00e4t W\u00fcrzburg]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1007\/s00025-018-0908-4","article-title":"Some Dynamic Inequalities Involving Hilbert and Hardy\u2013Hilbert Operators with Kernels","volume":"73","author":"Rezk","year":"2018","journal-title":"Results Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.7153\/mia-2017-20-65","article-title":"New Hilbert\u2019s dynamic inequalities on time scales","volume":"20","author":"Saker","year":"2017","journal-title":"Math. Inequalities Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1186\/s13660-019-2255-8","article-title":"Refinement multidimensional dynamic inequalities with general kernels and measures","volume":"2019","author":"Saker","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_20","first-page":"347","article-title":"A variety of inverse Hilbert type inequality on time scales","volume":"24","author":"Saker","year":"2017","journal-title":"Dyn. Contin. Discret. Impuls. Syst. Ser. A Math. Anal."},{"key":"ref_21","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":"113","author":"Saker","year":"2019","journal-title":"Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A Mat."},{"key":"ref_22","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_23","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_24","doi-asserted-by":"crossref","unstructured":"Agarwal, R., O\u2019Regan, D., and Saker, S.H. (2014). Dynamic Inequalities on Time Scales, Springer International Publishing.","DOI":"10.1007\/978-3-319-11002-8"},{"key":"ref_25","first-page":"477","article-title":"Hardy-Knopp-type inequalities on time scales","volume":"17","author":"Ozkan","year":"2008","journal-title":"Dyn. Syst. Appl."},{"key":"ref_26","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. Inequalities"},{"key":"ref_27","unstructured":"Mitrinovic, D.S., Pecaric, J.E., and Fink, A.M. (1993). Classical and New Inequalities in Analysis, Kluwer Academic."},{"key":"ref_28","first-page":"4976050","article-title":"Some Dynamic Inequalities of Hilbert\u2019s Type","volume":"2020","author":"Ahmed","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"399","DOI":"10.2298\/AADM170428001S","article-title":"On Hilbert\u2019s inequality on time scales","volume":"11","author":"Saker","year":"2017","journal-title":"Appl. Anal. Discret. 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