{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T10:55:10Z","timestamp":1769338510492,"version":"3.49.0"},"reference-count":50,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T00:00:00Z","timestamp":1703721600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science, Research and Innovation Fund (NSRF), and Suan Dusit"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Within the recent wave of research advancements, mathematical inequalities and their practical applications play a notably significant role across various domains. In this regard, inequalities offer a captivating arena for scholarly endeavors and investigational pursuits. This research work aims to present new improvements for the integral majorization inequalities using an interesting aproach. Certain previous improvements have been achieved for the Jensen inequality as direct outcomes of the main results. Additionally, estimates for the Csisz\u00e1r divergence and its cases are provided as applications of the main results. The circumstances under which the principal outcomes offer enhanced estimations for majorization differences are also underscored and emphasized.<\/jats:p>","DOI":"10.3390\/axioms13010021","type":"journal-article","created":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T11:44:57Z","timestamp":1703763897000},"page":"21","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Improvements of Integral Majorization Inequality with Applications to Divergences"],"prefix":"10.3390","volume":"13","author":[{"given":"Abdul","family":"Basir","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5373-4663","authenticated-orcid":false,"given":"Muhammad Adil","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan"}]},{"given":"Hidayat","family":"Ullah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1149-9559","authenticated-orcid":false,"given":"Yahya","family":"Almalki","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia"}]},{"given":"Chanisara","family":"Metpattarahiran","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1402","authenticated-orcid":false,"given":"Thanin","family":"Sitthiwirattham","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"},{"name":"Research Group for Fractional Calculus Theory and Applications, Science and Technology Research Institute, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,28]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H., Littlewood, J.E., and P\u00f3lya, G. 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