{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T13:23:24Z","timestamp":1780579404311,"version":"3.54.1"},"reference-count":55,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,7]],"date-time":"2020-04-07T00:00:00Z","timestamp":1586217600000},"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>Our work is based on the multiple inequalities illustrated in 2020 by Hamiaz and Abuelela. With the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalize a number of those inequalities to a general time scale. Besides that, in order to get new results as special cases, we will extend our results to continuous and discrete calculus.<\/jats:p>","DOI":"10.3390\/sym12040582","type":"journal-article","created":{"date-parts":[[2020,4,9]],"date-time":"2020-04-09T14:42:03Z","timestamp":1586443323000},"page":"582","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform"],"prefix":"10.3390","volume":"12","author":[{"given":"Ahmed A.","family":"El-Deeb","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Samer D.","family":"Makharesh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey"},{"name":"Institute of Space Science, 07650 Magurele-Bucharest, Romania"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13660-020-02310-2","article-title":"Some new discrete Hilbert\u2019s inequalities involving Fenchel\u2013Legendre transform","volume":"2020","author":"Hamiaz","year":"2020","journal-title":"J. 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