{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T20:30:23Z","timestamp":1770064223722,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,18]],"date-time":"2022-10-18T00:00:00Z","timestamp":1666051200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we prove several new fractional nabla Bennett\u2013Leindler dynamic inequalities with the help of a simple consequence of Keller\u2019s chain rule, integration by parts, mean inequalities and H\u00f6lder\u2019s inequality for the nabla fractional derivative on time scales. As a result of this, some new classical inequalities are obtained as special cases, and we extended our inequalities to discrete and continuous calculus. In addition, when \u03b1=1, we shall obtain some well-known dynamic inequalities as special instances from our results. Symmetrical properties are critical in determining the best ways to solve inequalities.<\/jats:p>","DOI":"10.3390\/sym14102183","type":"journal-article","created":{"date-parts":[[2022,10,19]],"date-time":"2022-10-19T02:54:25Z","timestamp":1666148065000},"page":"2183","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Some New Bennett\u2013Leindler Type Inequalities via Conformable Fractional Nabla Calculus"],"prefix":"10.3390","volume":"14","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"}]},{"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"}]},{"given":"Roqia","family":"Butush","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Jordan, Amman P.O. Box 11941, Jordan"}]},{"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":[[2022,10,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1007\/BF01199965","article-title":"Notes on a Theorem of Hilbert","volume":"6","author":"Hardy","year":"1920","journal-title":"Math. Z."},{"key":"ref_2","first-page":"150","article-title":"Notes on Some Points in the Integral Calculus, LX. An Inequality Between Integrals","volume":"54","author":"Hardy","year":"1925","journal-title":"Mess. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1112\/jlms\/s1-3.1.49","article-title":"Note on Series of Positive Terms","volume":"3","author":"Copson","year":"1928","journal-title":"J. Lond.Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1017\/S0308210500017868","article-title":"Some Integral Inequalities","volume":"75","author":"Copson","year":"1976","journal-title":"Prof. Roy. Soc. Edinburg. Sect. A"},{"key":"ref_5","first-page":"297","article-title":"Generalization of Inequalities of Hardy and Littlewood","volume":"31","author":"Leindler","year":"1970","journal-title":"Acta Sci. Math. (Szeged)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1093\/qmath\/38.4.401","article-title":"Some Elementary Inequalities","volume":"38","author":"Bennett","year":"1987","journal-title":"Quart. J. Math. Oxf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1093\/qmath\/39.4.385","article-title":"Some Elementary Inequalities II","volume":"2","author":"Bennett","year":"1988","journal-title":"Quart. J. Math."},{"key":"ref_8","first-page":"261","article-title":"Some Inequalities Pertaining to Bennetts Results","volume":"58","author":"Leindler","year":"1993","journal-title":"Acta Sci. Math. (Szeged)"},{"key":"ref_9","unstructured":"Hilger, S. (1988). Ein Ma\u00dfkettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten. [Ph.D. Dissertation, Universitat of W\u00fcrzburg]."},{"key":"ref_10","first-page":"495","article-title":"Hardy inequality on time scales and its application to half-linear dynamic equations","volume":"5","year":"2005","journal-title":"J. Inequal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1002\/mana.201300010","article-title":"Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales","volume":"287","author":"Saker","year":"2014","journal-title":"Math. Nachr."},{"key":"ref_12","first-page":"1","article-title":"Bennett-Leindler Type Inequalities for Nabla Time Scale Calculus","volume":"4","author":"Kayar","year":"2021","journal-title":"Mediterr. J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zakarya, M., Altanji, M., AlNemer, G., Abd El-Hamid, H.A., Cesarano, C., and Rezk, H.M. (2021). Fractional Reverse Coposn\u2019s Inequalities via Conformable Calculus on Time Scales. Symmetry, 13.","DOI":"10.3390\/sym13040542"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2050283","DOI":"10.1142\/S0217979220502835","article-title":"Conserved quantities along with Painlev\u00e9 analysis and optical solitons for the nonlinear dynamics of Heisenberg ferromagnetic spin chains model","volume":"34","author":"Ali","year":"2020","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"AlNemer, G., Kenawy, M.R., Zakarya, M., Cesarano, C., and Rezk, H.M. (2021). Generalizations of Hardy\u2019s Type Inequalities via Conformable Calculus. Symmetry, 13.","DOI":"10.3390\/sym13020242"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4094","DOI":"10.1002\/mma.7013","article-title":"Dispersive of propagation wave solutions to unidirectional shallow water wave Dullin\u2013Gottwald\u2013Holm system and modulation instability analysis","volume":"44","author":"Bilal","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/s12043-019-1843-y","article-title":"Tsallis holographic dark energy in Bianchi-I Universe using hybrid expansion law with k-essence","volume":"93","author":"Dubey","year":"2019","journal-title":"Pramana-J. Phys."},{"key":"ref_18","first-page":"1","article-title":"Hilbert-Type Inequalities for Time Scale Nabla Calculus","volume":"619","author":"AlNemer","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","first-page":"2399182","article-title":"Hardy-Leindler Type Inequalities via Conformable Delta Fractional Calculus","volume":"2022","author":"Rezk","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"103661","DOI":"10.1016\/j.rinp.2020.103661","article-title":"Lump and Interaction solutions of a geophysical Korteweg\u2013de Vries equation","volume":"19","author":"Rizvi","year":"2020","journal-title":"Results Phys."},{"key":"ref_21","first-page":"241","article-title":"Converses of Copson\u2019s Inequalities on Time Scales","volume":"18","author":"Saker","year":"2015","journal-title":"J. Math. Equal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"102775","DOI":"10.1016\/j.rinp.2019.102775","article-title":"Analytical wave solutions of the (2+1)-dimensional first integro-di erential Kadomtsev-Petviashivili hierarchy equation by using modified mathematical methods","volume":"15","author":"Seadawy","year":"2019","journal-title":"Results Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/s12043-019-1771-x","article-title":"Application of mathematical methods on the system of dynamical equations for the ion sound and Langmuir waves","volume":"93","author":"Seadawy","year":"2019","journal-title":"Pramana-J. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.jksus.2015.05.003","article-title":"A Conformable Fractional Calculus on Arbitrary Time Scales","volume":"28","author":"Benkhettou","year":"2016","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s40065-016-0160-2","article-title":"Chain Rules and Inequalities for the BHT Fractional Calculus on Arbitrary Times Sales","volume":"6","author":"Nwaeze","year":"2017","journal-title":"Arab J. Math."},{"key":"ref_26","first-page":"202","article-title":"A nabla conformable fractional calculus on time scales","volume":"7","author":"Bendouma","year":"2019","journal-title":"Math. Analy. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2183\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:09Z","timestamp":1760144169000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2183"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,18]]},"references-count":26,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102183"],"URL":"https:\/\/doi.org\/10.3390\/sym14102183","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,18]]}}}