{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T19:35:27Z","timestamp":1775590527389,"version":"3.50.1"},"reference-count":17,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T00:00:00Z","timestamp":1767830400000},"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\/248\/46"],"award-info":[{"award-number":["RGP 2\/248\/46"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work presents refined and generalized forms of weighted Opial-type inequalities within the framework of time scale calculus. The proofs rely on several algebraic techniques, together with H\u00f6lder\u2019s inequality and Keller\u2019s chain rule. These results extend the classical Opial-type inequalities by embedding them into the time scale setting, which unifies both continuous and discrete analyses. Consequently, various integral and discrete inequalities emerge as particular cases of our main results, thereby broadening the applicability of Opial-type inequalities to dynamic systems and discrete models.<\/jats:p>","DOI":"10.3390\/axioms15010046","type":"journal-article","created":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T14:32:38Z","timestamp":1767882758000},"page":"46","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Advanced Generalizations of Weighted Opial-Type Inequalities in the Framework of Time Scale Calculus"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-7968-4913","authenticated-orcid":false,"given":"Nadiah Zafer","family":"Al-Shehri","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"}]},{"given":"Mohammed M. A.","family":"El-Sheikh","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4312-8330","authenticated-orcid":false,"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0159-6726","authenticated-orcid":false,"given":"Hegagi M.","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, University of Bisha, Bisha 61922, 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"}]},{"given":"Fatma M.","family":"Khamis","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Women for (Art, Science, and Education), Ain Shams University, Cairo 11566, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D.S., and Vasi\u0107, P.M. (1970). Analytic Inequalities, Springer.","DOI":"10.1007\/978-3-642-99970-3"},{"key":"ref_2","unstructured":"Milovanovi\u0107, G.V., and Milovanovi\u0107, I.\u017d. (2025, November 04). The Best Constant in Some Integral Inequalities of Opial Type. Available online: https:\/\/www.researchgate.net\/publication\/235141373_The_best_constant_in_some_integral_inequalities_of_Opial_type."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Opial, Z. (2025, November 04). Sur une int\u00e9galit\u00e9, Annales Polonici Mathematici. 1960, 8, 29\u201332. Available online: http:\/\/eudml.org\/doc\/208439.","DOI":"10.4064\/ap-8-1-29-32"},{"key":"ref_4","first-page":"789","article-title":"On an inequality of opial","volume":"14","author":"Hua","year":"1965","journal-title":"Sci. Sin."},{"key":"ref_5","first-page":"78","article-title":"On a certain result of Z. Opial","volume":"42","author":"Yang","year":"1966","journal-title":"Proc. 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Hardy Type Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-44299-0"},{"key":"ref_14","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_15","doi-asserted-by":"crossref","unstructured":"Agarwal, R., O\u2019Regan, D., and Saker, S.H. (2014). Dynamic Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-11002-8"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"AlNemer, G., Saied, A.I., Hassan, A.M., Cesarano, C., Rezk, H.M., and Zakarya, M. (2022). On some new dynamic inequalities involving C-monotonic functions on time scales. Axioms, 11.","DOI":"10.3390\/axioms11110644"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., and Pang, P.Y.H. (1995). Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8426-5"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T14:36:08Z","timestamp":1767882968000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/1\/46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,8]]},"references-count":17,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["axioms15010046"],"URL":"https:\/\/doi.org\/10.3390\/axioms15010046","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,8]]}}}