{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T13:31:00Z","timestamp":1769520660172,"version":"3.49.0"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T00:00:00Z","timestamp":1647216000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007345","name":"King Mongkut's University of Technology North Bangkok","doi-asserted-by":"publisher","award":["KMUTNB-63-KNOW-20"],"award-info":[{"award-number":["KMUTNB-63-KNOW-20"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some Ostrowski\u2013Mercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of the Ostrowski inequalities established inside the literature. There are also some applications to the special means of real numbers given.<\/jats:p>","DOI":"10.3390\/axioms11030132","type":"journal-article","created":{"date-parts":[[2022,3,15]],"date-time":"2022-03-15T03:06:10Z","timestamp":1647313570000},"page":"132","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["On Some New Ostrowski\u2013Mercer-Type Inequalities for Differentiable Functions"],"prefix":"10.3390","volume":"11","author":[{"given":"Ifra Bashir","family":"Sial","sequence":"first","affiliation":[{"name":"School of Mathematics Science, Jiangsu University, Zhenjiang 212114, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nichaphat","family":"Patanarapeelert","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5341-4926","authenticated-orcid":false,"given":"Muhammad Aamir","family":"Ali","sequence":"additional","affiliation":[{"name":"Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China"}],"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 86120, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1007\/BF01214290","article-title":"\u00dcber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert","volume":"10","author":"Ostrowski","year":"1938","journal-title":"Comment. Math. Helv."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Budak, H., Sarikaya, M.Z., and Dragomir, S.S. (2018). Some perturbed Ostrowski type inequalities for twice differentiable functions. Advances in Mathematical Inequalities and Applications, Birkh\u00e4user.","DOI":"10.1007\/978-981-13-3013-1_14"},{"key":"ref_3","first-page":"1","article-title":"An inequality of Ostrowski type for mappings whose second derivatives are bounded and applications","volume":"15","author":"Cerone","year":"1999","journal-title":"East Asian Math. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/S0898-1221(99)00282-5","article-title":"The Ostrowski integral inequality for Lipschitzian mappings and applications","volume":"38","author":"Dragomir","year":"1999","journal-title":"Comput. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1090\/proc\/13488","article-title":"Generalized Ostrowski type inequalities for local fractional integrals","volume":"145","author":"Sarikaya","year":"2017","journal-title":"Proc. The American Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1016\/j.camwa.2011.12.023","article-title":"New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals","volume":"63","author":"Set","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D.S., Pexcxarixcx, J.E., and Fink, A.M. (1993). Classical and New Inequalities in Analysis, Kluwer Academic Publishers Group.","DOI":"10.1007\/978-94-017-1043-5"},{"key":"ref_8","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2000). Selected Topics on Hermite-Hadamard Inequalities and Applications, Victoria University. RGMIA Monographs."},{"key":"ref_9","unstructured":"Pe\u0107arixcx, J.E., Proschan, F., and Tong, Y.L. (1992). Convex Functions, Partial Orderings and Statistical Applications, Academic Press."},{"key":"ref_10","first-page":"Art 73","article-title":"A Variant of Jensen\u00eds Inequality","volume":"4","author":"Mercer","year":"2003","journal-title":"J. Ineq. Pure and Appl. Math"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"742","DOI":"10.13001\/1081-3810.1684","article-title":"Refinements of the operator Jensen-Mercer inequality","volume":"26","author":"Kian","year":"2013","journal-title":"Electron. J. Linear Algebra"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"712","DOI":"10.3934\/math.2021043","article-title":"On inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional integrals","volume":"6","author":"Abdeljawad","year":"2021","journal-title":"AIMS Math."},{"key":"ref_13","first-page":"60","article-title":"Generalized integral Mercer\u2019s inequality and integral means","volume":"10","author":"Ali","year":"2019","journal-title":"J. Inequal. Spec. Funct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3079","DOI":"10.1016\/j.aej.2020.06.040","article-title":"New fractional estimates for Hermite-Hadamard-Mercer\u2019s type inequalities","volume":"59","author":"Chu","year":"2020","journal-title":"Alexadria Eng. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/j.na.2009.01.120","article-title":"A generalization of Mercer\u2019s result on convex functions","volume":"71","author":"Niezgoda","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_16","first-page":"5516987","article-title":"The Hermite\u2013Hadamard-Jensen-Mercer Type Inequalities for Riemann-Liouville Fractional Integral","volume":"2021","author":"Wang","year":"2021","journal-title":"J. Math."},{"key":"ref_17","first-page":"5868326","article-title":"New Fractional Hermite\u2013Hadamard\u2013Mercer Inequalities for Harmonically Convex Function","volume":"2021","author":"Butt","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Vivas-Cortez, M., Saleem, M.S., Sajid, S., Zahoor, M.S., and Kashuri, A. (2021). Hermite-Jensen-Mercer-Type Inequalities via Caputo-Fabrizio Fractional Integral for h-Convex Function. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040269"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Set, E., Celik, B., Ozdemir, M.E., and Aslan, M. (2021). Some New Results on Hermite\u2013Hadamard-Mercer-Type Inequalities Using a General Family of Fractional Integral Operators. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030068"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1515\/dema-2004-0208","article-title":"Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions","volume":"37","author":"Cerone","year":"2004","journal-title":"Demonstratio Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1016\/j.aml.2010.04.038","article-title":"Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense","volume":"23","author":"Alomari","year":"2010","journal-title":"Appl. Math. Lett."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/3\/132\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:36:18Z","timestamp":1760135778000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/3\/132"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,14]]},"references-count":21,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["axioms11030132"],"URL":"https:\/\/doi.org\/10.3390\/axioms11030132","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,14]]}}}