{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,26]],"date-time":"2026-04-26T00:51:02Z","timestamp":1777164662290,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T00:00:00Z","timestamp":1694908800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Khalid University","award":["RGP.2\/102\/44"],"award-info":[{"award-number":["RGP.2\/102\/44"]}]},{"name":"King Khalid University","award":["PNURSP2023R337"],"award-info":[{"award-number":["PNURSP2023R337"]}]},{"name":"Princess Nourah bint Abdulrahman University Researchers","award":["RGP.2\/102\/44"],"award-info":[{"award-number":["RGP.2\/102\/44"]}]},{"name":"Princess Nourah bint Abdulrahman University Researchers","award":["PNURSP2023R337"],"award-info":[{"award-number":["PNURSP2023R337"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The primary intent of this study is to establish some important inequalities of the Hermite\u2013Hadamard, trapezoid, and midpoint types under fractional extended Riemann\u2013Liouville integrals (FERLIs). The proofs are constructed using the renowned Jensen\u2013Mercer, power-mean, and Holder inequalities. Various equalities for the FERLIs and convex functions are construed to be the mainstay for finding new results. Some connections between our main findings and previous research on Riemann\u2013Liouville fractional integrals and FERLIs are also discussed. Moreover, a number of examples are featured, with graphical representations to illustrate and validate the accuracy of the new findings.<\/jats:p>","DOI":"10.3390\/axioms12090886","type":"journal-article","created":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T23:57:46Z","timestamp":1694995066000},"page":"886","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Improvement in Some Inequalities via Jensen\u2013Mercer Inequality and Fractional Extended Riemann\u2013Liouville Integrals"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9273-9512","authenticated-orcid":false,"given":"Abd-Allah","family":"Hyder","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia"},{"name":"Department of Engineering Mathematics and Physics, Faculty of Engineering, Al-Azhar University, Cairo 71524, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7041-3730","authenticated-orcid":false,"given":"Areej A.","family":"Almoneef","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8843-955X","authenticated-orcid":false,"given":"H\u00fcseyin","family":"Budak","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, D\u00fczce University, D\u00fczce 81620, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF01170410","article-title":"Nonconvex energy functions. Hemivariational inequalities and substationarity principles","volume":"48","author":"Panagiotopoulos","year":"1983","journal-title":"Acta Mech."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Panagiotopoulos, P.D. (1985). 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