{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T21:52:05Z","timestamp":1770069125151,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T00:00:00Z","timestamp":1712016000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Khalid University","award":["RGP 2\/135\/44"],"award-info":[{"award-number":["RGP 2\/135\/44"]}]},{"name":"King Khalid University","award":["PNURSP2024R45"],"award-info":[{"award-number":["PNURSP2024R45"]}]},{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["RGP 2\/135\/44"],"award-info":[{"award-number":["RGP 2\/135\/44"]}]},{"name":"Princess Nourah bint Abdulrahman University Researchers Supporting Project","award":["PNURSP2024R45"],"award-info":[{"award-number":["PNURSP2024R45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we discuss several novel generalized Ostrowski-type inequalities for functions whose derivative module is relatively convex in time scales calculus. Our core findings are proved by using the integration by parts technique, H\u00f6lder\u2019s inequality, and the chain rule on time scales. These derived inequalities expand the existing literature, enriching specific integral inequalities within this domain.<\/jats:p>","DOI":"10.3390\/axioms13040235","type":"journal-article","created":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T10:58:31Z","timestamp":1712055511000},"page":"235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Inequalities of Ostrowski Type for Functions Whose Derivative Module Is Relatively Convex on Time Scales"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6782-7908","authenticated-orcid":false,"given":"Haytham M.","family":"Rezk","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}]},{"given":"Ahmed I.","family":"Saied","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"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\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"AlNemer","sequence":"additional","affiliation":[{"name":"Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"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, Sarat Ubaidah, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1007\/BF01214290","article-title":"\u0170ber die Absolutabweichung einer diferentiebaren Funktion von ihrem Integralmittelwert","volume":"10","author":"Ostrowski","year":"1938","journal-title":"Comment. Math. Helv."},{"key":"ref_2","first-page":"3","article-title":"Ostrowski type inequalities for quasi-convex functions with applications to special means","volume":"13","author":"Alomari","year":"2010","journal-title":"RGMIA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1016\/j.aml.2010.04.038","article-title":"Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense","volume":"23","author":"Alomari","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"121","DOI":"10.18576\/amis\/130116","article-title":"Ostrowski-type inequalities for functions whose derivative modulus is relatively convex","volume":"13","author":"Cortez","year":"2019","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_5","first-page":"2691","article-title":"More accurate dynamic Hardy-type inequalities obtained via superquadraticity","volume":"113","author":"Saker","year":"2019","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00eds. Nat. Ser. Mat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7668860","DOI":"10.1155\/2022\/7668860","article-title":"On a Refinement of Multidimensional Inequalities of Hardy-Type via Superquadratic and Subquadratic Functions","volume":"2022","author":"Zakarya","year":"2022","journal-title":"J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., O\u2019Regan, D., and Saker, S.H. (2016). Hardy Type Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-44299-0"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., O\u2019Regan, D., and Saker, S.H. (2014). Dynamic Inequalities on Time Scales, Springer.","DOI":"10.1007\/978-3-319-11002-8"},{"key":"ref_9","first-page":"1","article-title":"Nabla dynamic equations on time scales","volume":"13","author":"Anderson","year":"2003","journal-title":"Panam. Math. J."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Georgiev, S.G. (2016). Multiple Integration on Time Scales. Multivariable Dynamic Calculus on Time Scales, Springer.","DOI":"10.1007\/978-3-319-47620-9"},{"key":"ref_12","first-page":"903196","article-title":"Hardy-type inequalities on time scales vie convexity in several variables","volume":"2013","author":"Donchev","year":"2013","journal-title":"Int. Sch. Res. Not."},{"key":"ref_13","first-page":"87","article-title":"Convex functions on time scales","volume":"35","author":"Dinu","year":"2008","journal-title":"Ann. Univ. Craiova-Math. Comput. Sci. Ser."},{"key":"ref_14","unstructured":"Hilger, S. (1988). Ein Makettenkalk\u00fcl mit Anwendung auf Zentrmsmannigfaltingkeiten. [Ph.D. Thesis, University W\u00fcrzburg]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/BF03323153","article-title":"Analysis on measure chains\u2014A unified approach to continuous and discrete calculus","volume":"18","author":"Hilger","year":"1990","journal-title":"Results Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"61","DOI":"10.15352\/afa\/1396833503","article-title":"Time scales Hardy-type inequalities via superquadracity","volume":"5","author":"Oguntuase","year":"2014","journal-title":"Ann. Funct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1016\/j.aml.2007.06.008","article-title":"Extensions of certain integral inequalities on time scales","volume":"21","author":"Sarikaya","year":"2008","journal-title":"Appl. Math. Lett."},{"key":"ref_18","first-page":"495","article-title":"Hardy inequality on time scales and its application to half-linear dynamic equations","volume":"2005","year":"2005","journal-title":"J. Inequal. Appl."},{"key":"ref_19","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. Inequal."},{"key":"ref_20","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_21","first-page":"633","article-title":"The Hadamard\u2019s inequality for s-convex functions in the first sense","volume":"31","author":"Dragomir","year":"1998","journal-title":"Demonstr. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/235\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:22:27Z","timestamp":1760106147000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/4\/235"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,2]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["axioms13040235"],"URL":"https:\/\/doi.org\/10.3390\/axioms13040235","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,2]]}}}