{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,7]],"date-time":"2026-05-07T05:25:54Z","timestamp":1778131554042,"version":"3.51.4"},"reference-count":24,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,14]],"date-time":"2024-07-14T00:00:00Z","timestamp":1720915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Research and Graduate Studies at King Khalid University","award":["RGP 2\/190\/45"],"award-info":[{"award-number":["RGP 2\/190\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales T. We achieve this by utilizing H\u00f6lder\u2019s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when T=N and T=R), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as T=qN0 for q&gt;1, which, to the best of the authors\u2019 knowledge, is a largely novel conclusion.<\/jats:p>","DOI":"10.3390\/axioms13070475","type":"journal-article","created":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T14:15:49Z","timestamp":1721052949000},"page":"475","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Some New Dynamic Hilbert-Type Inequalities across Time Scales"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4312-8330","authenticated-orcid":false,"given":"Mohammed","family":"Zakarya","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"}]},{"given":"Ahmed I.","family":"Saied","sequence":"additional","affiliation":[{"name":"Mathematical Institute, Slovak Academy of Sciences, Greko\u0161\u00e1kova 6, 04001 Ko\u0161ice, Slovakia"},{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"given":"Amirah Ayidh I","family":"Al-Thaqfan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, King Khalid University, P.O. Box 64512, Abha 62529, Sarat Ubaidah, Saudi Arabia"}]},{"given":"Maha","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Arts and Sciences, King Khalid University, P.O. Box 64512, Abha 62529, Sarat Ubaidah, Saudi Arabia"}]},{"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":[[2024,7,14]]},"reference":[{"key":"ref_1","unstructured":"Hilbert, D. (1906). Grundz\u00fcge Einer Allgemeinen Theorie der Linearen Intergraleichungen, Springer."},{"key":"ref_2","first-page":"1","article-title":"Bernerkungen sur theorie der beschrankten Bilinearformen mit unendlich vielen veranderlichen","volume":"140","author":"Schur","year":"1911","journal-title":"J. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1186\/s13660-017-1360-9","article-title":"A Hilbert-type fractal integral inequality and its applications","volume":"2017","author":"Liu","year":"2017","journal-title":"J. Inequal. 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