{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T08:19:04Z","timestamp":1769156344611,"version":"3.49.0"},"reference-count":39,"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":"King Saud University, Riyadh, Saudi Arabia","award":["RSP-2021\/158"],"award-info":[{"award-number":["RSP-2021\/158"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main objective of this paper is to establish some new variants of the Jensen\u2013Mercer inequality via harmonically strongly convex function. We also propose some new fractional analogues of Hermite\u2013Hadamard\u2013Jensen\u2013Mercer-like inequalities using AB fractional integrals. In order to obtain some of our main results, we also derive new fractional integral identities. To demonstrate the significance of our main results, we present some interesting applications to special means and to error bounds as well.<\/jats:p>","DOI":"10.3390\/sym14102187","type":"journal-article","created":{"date-parts":[[2022,10,19]],"date-time":"2022-10-19T02:54:25Z","timestamp":1666148065000},"page":"2187","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Jensen-Mercer Type Inequalities in the Setting of Fractional Calculus with Applications"],"prefix":"10.3390","volume":"14","author":[{"given":"Bandar","family":"Bin-Mohsin","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5212-6252","authenticated-orcid":false,"given":"Muhammad Zakria","family":"Javed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1019-9485","authenticated-orcid":false,"given":"Muhammad Uzair","family":"Awan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcela V.","family":"Mihai","sequence":"additional","affiliation":[{"name":"Department Scientific-Methodical Sessions, Romanian Mathematical Society-Branch Bucharest, Academy Street no. 14, RO-010014 Bucharest, Romania"}],"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, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Awais Gul","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6105-2435","authenticated-orcid":false,"given":"Muhammad Aslam","family":"Noor","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 4000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1016\/j.jmaa.2005.04.048","article-title":"Convexity with respect to a differential equation","volume":"315","author":"Ferreira","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","first-page":"345","article-title":"Centrally symmetric convex sets","volume":"14","author":"Boltyanski","year":"2007","journal-title":"J. Convex Anal."},{"key":"ref_3","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2000). Selected Topics on Hermite-Hadamard Inequalities and Applications. Victoria University."},{"key":"ref_4","first-page":"73","article-title":"A variant of Jensen\u2019s inequality","volume":"4","author":"Mercer","year":"2003","journal-title":"J. Inequal Pure Appl. Math."},{"key":"ref_5","first-page":"935","article-title":"Hermite-Hadamard type inequalities for harmonically convex functions","volume":"43","author":"Iscan","year":"2014","journal-title":"Hacettepe J. Math. Stat."},{"key":"ref_6","first-page":"1","article-title":"Asian options with harmonic average","volume":"9","author":"Calin","year":"2015","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_7","unstructured":"Noor, M.A. (2017). Advanced Convex Analysis and Optimization, COMSATS University Islamabad. Lecture Notes."},{"key":"ref_8","first-page":"72","article-title":"Existence theorems and convergence of minimizing sequences in extremum problems with restrictions","volume":"7","author":"Polyak","year":"1966","journal-title":"Sov. Math. Dokl."},{"key":"ref_9","first-page":"66","article-title":"Strongly generalized harmonic convex functions and integral inequalities","volume":"7","author":"Noor","year":"2016","journal-title":"J. Math. Anal."},{"key":"ref_10","first-page":"99","article-title":"Hermite-Hadamard inequalities for strongly harmonic convex functions","volume":"7","author":"Noor","year":"2016","journal-title":"J. Inequal. Spec. Funct."},{"key":"ref_11","first-page":"427","article-title":"Some properties of generalized strongly harmonic convex functions","volume":"16","author":"Noor","year":"2018","journal-title":"Int. J. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3783","DOI":"10.2298\/FIL1912783A","article-title":"On approximately harmonic h-convex functions depending on a given function","volume":"33","author":"Awan","year":"2019","journal-title":"Filomat"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6404","DOI":"10.3934\/math.2020412","article-title":"A varient of Jensen-type inequality and related results for harmonic convex functions","volume":"5","author":"Baloch","year":"2020","journal-title":"Aims Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s00010-017-0496-5","article-title":"Around Jensen\u2019s inequality for strongly convex functions","volume":"92","author":"Moradi","year":"2017","journal-title":"Aequationes Math."},{"key":"ref_15","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Sci. B.V.. North-Holland Mathematics Studies 204."},{"key":"ref_16","first-page":"191","article-title":"On generalized Wright\u2019s hypergeometric functions and fractional calculus operators","volume":"21","author":"Raina","year":"2005","journal-title":"East Asian Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.cam.2018.12.030","article-title":"Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals","volume":"353","author":"Ahmad","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_18","first-page":"73","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/S0034-4877(17)30059-9","article-title":"On Fractional Derivatives with Exponential Kernel and their Discrete Versions","volume":"80","author":"Abdeljawad","year":"2017","journal-title":"Rep. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New Fractional Derivatices with Non-Local and Non-Singular Kernel: Theory and Application to Heat Transfer Model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1016\/j.mcm.2011.12.048","article-title":"Hermite-Hadamard\u2019s inequalities for fractional integrals and related fractional inequalities","volume":"57","author":"Sarikaya","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2425","DOI":"10.2298\/FIL2107425O","article-title":"Hermite-Hadamard-Mercer type inequalities for fractional integrals","volume":"35","author":"Ogulmus","year":"2021","journal-title":"Filomat"},{"key":"ref_23","first-page":"7061549","article-title":"Hermite-Jensen-Mercer type inequalities for Caputo fractional derivatives","volume":"2020","author":"Zhao","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_24","first-page":"1","article-title":"New Hermite-Jensen-Mercer-type inequalities via k-fractional integrals","volume":"2020","author":"Butt","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5193","DOI":"10.3934\/math.2020334","article-title":"Hermite-Jensen-Mercer type inequalities via \u03a8-Riemann-Liouville k-fractional integrals","volume":"5","author":"Butt","year":"2020","journal-title":"AIMS Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3203","DOI":"10.3934\/math.2022177","article-title":"Some new generalized \u03ba-fractional Hermite-Hadamard-Mercer type integral inequalities and their applications","volume":"7","author":"Cortez","year":"2022","journal-title":"AIMS Math."},{"key":"ref_27","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_28","first-page":"21","article-title":"Jensen-Mercer inequality for GA-convex functions and some related inequalities","volume":"2020","author":"Iscan","year":"2020","journal-title":"J. Inequalities Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1811","DOI":"10.18576\/amis\/100522","article-title":"Harmonic Variational Inequalities","volume":"10","author":"Noor","year":"2016","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"831","DOI":"10.18576\/amis\/120418","article-title":"Fractal Integral Inequalities for Harmonic Convex Functions","volume":"12","author":"Noor","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"589","DOI":"10.18576\/isl\/110227","article-title":"Quasi variational inclusions involving three operators","volume":"11","author":"Noor","year":"2022","journal-title":"Inform. Sci. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"405","DOI":"10.18576\/amis\/120215","article-title":"Hermite-Hadamard Inequalities for Exponentially Convex Functions","volume":"12","author":"Awan","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Faisal, S., Adil Khan, M., Khan, T.U., Saeed, T., Alshehri, A.M., and Nwaeze, E.R. (2022). New \u201cConticrete\u201d Hermite-Hadamard-Jensen-Mercer Fractional Inequalities. Symmetry, 14.","DOI":"10.3390\/sym14020294"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2040035","DOI":"10.1142\/S0218348X20400356","article-title":"On the new explicit solutions of the fractional nonlinear space-time nuclear model","volume":"28","author":"Khater","year":"2020","journal-title":"Fractals"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"109395","DOI":"10.1016\/j.chaos.2019.109395","article-title":"An analytical solution for fractional oscillator in a resisting medium","volume":"130","author":"Ismail","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"877","DOI":"10.3934\/dcdss.2019058","article-title":"An independent set degree condition for fractional critical deleted graphs","volume":"12","author":"Gao","year":"2019","journal-title":"Discret. Contin. Dyn. Syst.\u2014S"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.2298\/TSCI180320239Y","article-title":"A New General Fractional-Order Derivative with Rabotnov Fractional-Exponential Kernel Applied to Model the Anomalous Heat Transfer","volume":"23","author":"Yang","year":"2019","journal-title":"Therm. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"125","DOI":"10.12785\/isl\/030306","article-title":"Generalized Fractal Dimensions in Image Thresholding Technique","volume":"3","author":"Uthayakumar","year":"2014","journal-title":"Inf. Sci. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"217","DOI":"10.12785\/ijtfst\/020307","article-title":"New Design of Hairpin-Koch Fractal Filter for Suppression of Spurious Band","volume":"2","author":"Shaalan","year":"2013","journal-title":"Int. J. Thin Film Sci. Technol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2187\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:22Z","timestamp":1760144182000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2187"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,18]]},"references-count":39,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102187"],"URL":"https:\/\/doi.org\/10.3390\/sym14102187","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,18]]}}}