{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T14:06:00Z","timestamp":1773151560371,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T00:00:00Z","timestamp":1645401600000},"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>The main objective of the present article is to prove some new \u2207 dynamic inequalities of Hardy\u2013Hilbert-type on time scales. We present and prove very important generalized results with the help of the Fenchel\u2013Legendre transform, submultiplicative functions, and H\u00f6lder\u2019s and Jensen\u2019s inequality on time scales. We obtain some well-known time scale inequalities due to Hardy\u2013Hilbert inequalities. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Symmetry plays an essential role in determining the correct methods for solutions to dynamic inequalities<\/jats:p>","DOI":"10.3390\/sym14020428","type":"journal-article","created":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T20:48:41Z","timestamp":1645476521000},"page":"428","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Important Study on the \u2207 Dynamic Hardy\u2013Hilbert-Type Inequalities on Time Scales with Applications"],"prefix":"10.3390","volume":"14","author":[{"given":"Ahmed A.","family":"El-Deeb","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo 11884, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,21]]},"reference":[{"key":"ref_1","unstructured":"Hilger, S. 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