{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T08:48:13Z","timestamp":1762073293607,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,29]],"date-time":"2022-10-29T00:00:00Z","timestamp":1667001600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Fund of Khon Kaen University, Thailand"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the attention of several mathematicians because of its real-life applications. More importantly, it has turned into a valuable tool for handling elements from perplexing frameworks within different parts of the pure and applied sciences. Integral inequalities, in association with convexity, have a strong relationship with symmetry. The objective of this article is to introduce the notion of operator refined exponential type convexity. Fractional versions of the Hermite\u2013Hadamard type inequality employing generalized R\u2013L fractional operators are established. Additionally, some novel refinements of Hermite\u2013Hadamard type inequalities are also discussed using some established identities. Finally, we present some applications of the probability density function and special means of real numbers.<\/jats:p>","DOI":"10.3390\/axioms11110602","type":"journal-article","created":{"date-parts":[[2022,10,29]],"date-time":"2022-10-29T23:45:00Z","timestamp":1667087100000},"page":"602","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Some New Integral Inequalities Involving Fractional Operator with Applications to Probability Density Functions and Special Means"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2751-7793","authenticated-orcid":false,"given":"Bibhakar","family":"Kodamasingh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Institute of Technical Education and Research, Siksha \u2018O\u2019 Anusandhan University, Bhubaneswar 751030, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4524-1951","authenticated-orcid":false,"given":"Soubhagya Kumar","family":"Sahoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Institute of Technical Education and Research, Siksha \u2018O\u2019 Anusandhan University, Bhubaneswar 751030, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0269-7828","authenticated-orcid":false,"given":"Wajid Ali","family":"Shaikh","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Quest, Nawabshah 67450, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8372-2532","authenticated-orcid":false,"given":"Muhammad","family":"Tariq","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,29]]},"reference":[{"key":"ref_1","first-page":"2","article-title":"On some inequalities for s-convex functions and applications","volume":"333","author":"Yildiz","year":"2013","journal-title":"J. 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