{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:12:31Z","timestamp":1767917551686,"version":"3.49.0"},"reference-count":19,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T00:00:00Z","timestamp":1668470400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we establish some new dynamic inequalities involving C-monotonic functions with C\u22651, on time scales. As a special case of our results when C=1, we obtain the inequalities involving increasing or decreasing functions (where for C=1, the 1-decreasing function is decreasing and the 1-increasing function is increasing). The main results are proved by applying the properties of C-monotonic functions and the chain rule formula on time scales. As a special case of our results, when T=R, we obtain refinements of some well-known continuous inequalities and when T=N, to the best of the authors\u2019 knowledge, the results are essentially new.<\/jats:p>","DOI":"10.3390\/axioms11110644","type":"journal-article","created":{"date-parts":[[2022,11,16]],"date-time":"2022-11-16T02:36:36Z","timestamp":1668566196000},"page":"644","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On Some New Dynamic Inequalities Involving C-Monotonic Functions on Time Scales"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2222-7973","authenticated-orcid":false,"given":"Ghada","family":"AlNemer","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"A. I.","family":"Saied","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"given":"A. M.","family":"Hassan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, Universit\u00e0 Telematica Internazionale Uninettuno, 00186 Rome, Italy"}]},{"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"}]},{"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"},{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,15]]},"reference":[{"key":"ref_1","first-page":"133","article-title":"Weighted inequalities for monotone and concave functions","volume":"116","author":"Heinig","year":"1995","journal-title":"Stud. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1006\/jmaa.1997.5646","article-title":"Integral inequalities for monotone functions","volume":"215","author":"Perixcx","year":"1997","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"AlNemer, G., Saied, A.I., Zakarya, M., El-Hamid, H.A.A., Bazighifan, O., and Rezk, H.M. (2021). Some New Reverse Hilbert\u2019s Inequalities on Time Scales. Symmetry, 13.","DOI":"10.3390\/sym13122431"},{"key":"ref_4","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_5","first-page":"185","article-title":"Some dynamic Hardy-type inequalities with general kernels","volume":"8","author":"Bohner","year":"2014","journal-title":"Math. Ineq. Appl."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2013\/903196","article-title":"Hardy-type inequalities on time scales via convexity in several variables","volume":"2013","author":"Donchev","year":"2013","journal-title":"ISRN Math. Anal."},{"key":"ref_8","first-page":"29","article-title":"Multivariat Hardy\u2014Type inequalities on time scales via superquadraticity","volume":"167","author":"Fabelurin","year":"2015","journal-title":"Proc. A. Razm. Math. Inst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"444","DOI":"10.22436\/jnsa.011.04.01","article-title":"Some reverse H\u00f6lder inequalities with Specht\u2019s ratio on time scales","volume":"11","author":"Elsennary","year":"2018","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_10","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. Anal."},{"key":"ref_11","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. Inequalities Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Rezk, H.M., AlNemer, G., Saied, A.I., Bazighifan, O., and Zakarya, M. (2022). Some New Generalizations of Reverse Hilbert-Type Inequalities on Time Scales. Symmetry, 14.","DOI":"10.3390\/sym14040750"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2022\/7997299","article-title":"On Hardy\u2013Knopp Type Inequalities with Kernels via Time Scale Calculus","volume":"2022","author":"Rezk","year":"2022","journal-title":"J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s13398-020-00876-6","article-title":"Some new dynamic Hardy-type inequalities with kernels involving monotone functions","volume":"114","author":"Saker","year":"2020","journal-title":"Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat."},{"key":"ref_15","first-page":"1","article-title":"Some New Weighted Dynamic Inequalities for Monotone Functions Involving Kernels","volume":"17","author":"Saker","year":"2020","journal-title":"Mediterr. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1016\/j.aml.2004.06.028","article-title":"Young\u2019s inequality and related results on time scales","volume":"18","author":"Wong","year":"2005","journal-title":"Appl. Math. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2019\/7584836","article-title":"Some new dynamic inequalities involving monotonic functions on time scales","volume":"2019","author":"Saker","year":"2019","journal-title":"J. Funct. Spaces"},{"key":"ref_18","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_19","doi-asserted-by":"crossref","unstructured":"Bohner, M., and Peterson, A. (2002). Advances in Dynamic Equations on Time Scales, Springer Science & Business Media.","DOI":"10.1007\/978-0-8176-8230-9"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/644\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:18:19Z","timestamp":1760145499000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/644"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,15]]},"references-count":19,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["axioms11110644"],"URL":"https:\/\/doi.org\/10.3390\/axioms11110644","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,15]]}}}