{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T14:06:20Z","timestamp":1778681180793,"version":"3.51.4"},"reference-count":56,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T00:00:00Z","timestamp":1627689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>It is a familiar fact that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis, both the inclusion relation (\u2286) and pseudo order relation (\u2264p) are two different concepts. In this article, by using pseudo order relation, we introduce the new class of nonconvex functions known as LR-p-convex interval-valued functions (LR-p-convex-IVFs). With the help of this relation, we establish a strong relationship between LR-p-convex-IVFs and Hermite-Hadamard type inequalities (HH-type inequalities) via Katugampola fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-p-convex-IVFs and their variant forms as special cases. Useful examples that demonstrate the applicability of the theory proposed in this study are given. The concepts and techniques of this paper may be a starting point for further research in this area.<\/jats:p>","DOI":"10.3390\/axioms10030175","type":"journal-article","created":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T21:51:07Z","timestamp":1627854667000},"page":"175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":30,"title":["Some New Fractional Estimates of Inequalities for LR-p-Convex Interval-Valued Functions by Means of Pseudo Order Relation"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7450-8067","authenticated-orcid":false,"given":"Muhammad Bilal","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad Aslam","family":"Noor","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Institute of Space Sciences, P.O. Box MG-23, R 76900 Magurele-Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2788-809X","authenticated-orcid":false,"given":"Juan Luis Garc\u00eda","family":"Guirao","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada y Estad\u00edstica, Campus de la Muralla, Universidad Polit\u00e9cnica de Cartagena, 30203 Cartagena, Murcia, Spain"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,31]]},"reference":[{"key":"ref_1","first-page":"82","article-title":"Sur deux limites d\u2019une int\u00e9grale d\u00e9finie","volume":"3","author":"Hermite","year":"1883","journal-title":"Mathesis"},{"key":"ref_2","first-page":"171","article-title":"\u00c9tude sur les propri\u00e9t\u00e9s des fonctions enti\u00e8res et en particulier d\u2019une fonction consid\u00e9r\u00e9e par Riemann","volume":"7","author":"Hadamard","year":"1893","journal-title":"J. Math\u00e9matiques Pures Appliqu\u00e9es"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1186\/s13660-020-02393-x","article-title":"New Hermite\u2013Hadamard type inequalities for n-polynomial harmonically convex functions","volume":"2020","author":"Awan","year":"2020","journal-title":"J. Inequal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1186\/s13660-019-2272-7","article-title":"Hermite\u2013Hadamard type inequalities for co-ordinated convex and quasi-convex functions and their applications","volume":"2019","author":"Latif","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1002\/mana.200810197","article-title":"The Schur multiplicative and harmonic convexities of the complete symmetric function","volume":"284","author":"Chu","year":"2011","journal-title":"Math. Nachr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.jmva.2011.08.004","article-title":"The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications","volume":"105","author":"Chu","year":"2012","journal-title":"J. Multivar. Anal."},{"key":"ref_7","first-page":"9487823","article-title":"Integral majorization type inequalities for the functions in the sense of strong convexity","volume":"2019","author":"Khan","year":"2019","journal-title":"J. Funct. Spaces"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1186\/s13660-019-2007-9","article-title":"Majorization theorems for strongly convex functions","volume":"2019","author":"Chu","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.jmaa.2006.02.086","article-title":"On h-convexity","volume":"326","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","first-page":"130","article-title":"p-convex functions and their properties","volume":"23","author":"Zhang","year":"2007","journal-title":"Pure Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0165-0114(89)90268-6","article-title":"On variational inequalities for fuzzy mappings","volume":"32","author":"Chang","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/0165-0114(92)90256-4","article-title":"Convex fuzzy mappings","volume":"48","author":"Nanda","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0165-0114(94)90011-6","article-title":"Fuzzy preinvex functions","volume":"64","author":"Noor","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_14","first-page":"15","article-title":"A note on generalized convex functions","volume":"2019","author":"Chu","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"585","DOI":"10.18514\/MMN.2014.660","article-title":"New integral inequalities involving beta function via P-convexity","volume":"15","author":"Liu","year":"2014","journal-title":"Miskolc. Math. Notes"},{"key":"ref_16","first-page":"369","article-title":"Uberdie Fourierreihen II","volume":"24","year":"1906","journal-title":"Math. Naturwise. Anz. Ungar. Akad. Wiss."},{"key":"ref_17","first-page":"13","article-title":"Stetigkeitsaussagen f\u00fcr eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen R\u00e4umen","volume":"23","author":"Breckner","year":"1978","journal-title":"Pupl. Inst. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/BF01837981","article-title":"Some remarks on s-convex functions","volume":"48","author":"Hudzik","year":"1994","journal-title":"Aequat. Math."},{"key":"ref_19","first-page":"137","article-title":"Hermite\u2013Hadamard type inequalities for p-convex functions","volume":"11","author":"Iscan","year":"2016","journal-title":"Int. J. Anal Appl."},{"key":"ref_20","unstructured":"Moore, R.E. (1966). Interval Analysis, Prentice Hall."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.fss.2017.02.001","article-title":"Jensen\u2019s inequality type integral for fuzzy-interval-valued functions","volume":"327","author":"Costa","year":"2017","journal-title":"Fuzzy Sets Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.ins.2017.08.055","article-title":"Some integral inequalities for fuzzy-interval-valued functions","volume":"420","author":"Costa","year":"2017","journal-title":"Inform. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1306","DOI":"10.1007\/s40314-016-0396-7","article-title":"Some integral inequalities for interval-valued functions","volume":"37","author":"Lodwick","year":"2018","journal-title":"Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Roman-Flores, H., Chalco-Cano, Y., and Silva, G.N. (2013, January 24\u201328). A note on Gronwall type inequality for interval-valued functions. Proceedings of the IEEE IFSA World Congress and NAFIPS Annual Meeting, Edmonton, AB, Canada.","DOI":"10.1109\/IFSA-NAFIPS.2013.6608616"},{"key":"ref_25","first-page":"457","article-title":"Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative","volume":"31","year":"2012","journal-title":"Comput. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3293","DOI":"10.1007\/s00500-014-1483-6","article-title":"Ostrowski type inequalities and applications in numerical integration for interval-valued functions","volume":"19","author":"Lodwick","year":"2015","journal-title":"Soft Comput."},{"key":"ref_27","first-page":"979","article-title":"Jensen and Hermite\u2013Hadamard inequalities for strongly convex set-valued maps","volume":"4","author":"Nikodem","year":"2014","journal-title":"Math. Aterna"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1007\/BF03323058","article-title":"An integral Jensen inequality for convex multifunctions","volume":"26","author":"Matkowski","year":"1994","journal-title":"Results Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2019.06.002","article-title":"Jensen\u2019s inequalities for set-valued and fuzzy set-valued functions","volume":"2020","author":"Zhang","year":"2020","journal-title":"Fuzzy Sets Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1186\/s13662-017-1126-1","article-title":"Monotonicity results for fractional difference operators with discrete exponential kernels","volume":"2017","author":"Abdeljawad","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"7160","DOI":"10.1002\/mma.5822","article-title":"A mathematical fractional model with nonsingular kernel for thrombin receptor activation in calcium signaling","volume":"42","author":"Agarwal","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1062","DOI":"10.3934\/math.2020074","article-title":"Existence and uniqueness of miscible flow equation through porous media with a non-singular fractional derivative","volume":"5","author":"Agarwal","year":"2019","journal-title":"AIMS Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1051\/mmnp\/2018063","article-title":"A hybrid analytical algorithm for nonlinear fractional wave-like equations","volume":"14","author":"Kumar","year":"2019","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1090\/proc\/14741","article-title":"Fractional Hermite\u2013Hadamard type inequalities for interval-valued functions","volume":"148","author":"Budak","year":"2019","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_35","first-page":"1","article-title":"A new approach to generalized fractional derivatives","volume":"6","author":"Katugampola","year":"2014","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_36","first-page":"149","article-title":"Hermite\u2013Hadamard type inequalities for p-convex functions via Katugampola fractional integrals","volume":"34","author":"Toplu","year":"2019","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1186\/s13660-018-1664-4","article-title":"Ostrowski type inequalities involving conformable fractional integrals","volume":"2018","author":"Begum","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1186\/1029-242X-2014-45","article-title":"On the (p, h)-convex function and some integral inequalities","volume":"2014","author":"Fang","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.ajmsc.2016.11.001","article-title":"Hermite\u2013Hadamard\u2013Fej\u00e9r type inequalities for p-convex functions","volume":"23","author":"Kunt","year":"2017","journal-title":"Arab J. Math. Sci."},{"key":"ref_40","first-page":"26","article-title":"Fractional Hermite\u2013Hadamard type inequalities for interval-valued functions","volume":"2019","author":"Liu","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1016\/j.mcm.2011.12.048","article-title":"Hermite\u2013Hadamard\u2019s inequalities for fractional integrals and related fractional inequalities","volume":"57","author":"Sarikaya","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"896483","DOI":"10.1155\/2011\/896483","article-title":"Logarithmically complete monotonicity properties relating to the gamma function","volume":"2011","author":"Zhao","year":"2011","journal-title":"Abstr. Appl. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"4625","DOI":"10.1109\/TIM.2020.2983531","article-title":"Evolving fuzzy models for prosthetic hand myoelectric-based control","volume":"69","author":"Precup","year":"2020","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1186\/s13662-021-03245-8","article-title":"New Hermite-Hadamard Type Inequalities for (h1, h2)-Convex Fuzzy-Interval-Valued Functions","volume":"2021","author":"Khan","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_45","first-page":"1","article-title":"New Hermite\u2013Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense","volume":"2021","author":"Liu","year":"2021","journal-title":"Complex Intell. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.2991\/ijcis.d.210409.001","article-title":"Some New Classes of Preinvex Fuzzy-Interval-Valued Functions and Inequalities","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Mohammed, P.O., Noor, M.A., and Hamed, Y.S. (2021). New Hermite\u2013Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities. Symmetry, 13.","DOI":"10.3390\/sym13040673"},{"key":"ref_48","first-page":"6657602","article-title":"On Strongly Generalized Preinvex Fuzzy Mappings","volume":"2021","author":"Liu","year":"2021","journal-title":"J. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"6652930","DOI":"10.1155\/2021\/6652930","article-title":"Extended perturbed mixed variational-like inequalities for fuzzy mappings","volume":"2021","author":"Khan","year":"2021","journal-title":"J. Math."},{"key":"ref_50","first-page":"518","article-title":"Exponentially Preinvex Fuzzy Mappings and Fuzzy Exponentially Mixed Variational-Like Inequalities","volume":"19","author":"Khan","year":"2021","journal-title":"Int. J. Anal. Appl."},{"key":"ref_51","first-page":"459","article-title":"Some New Inequalities for LR-Log-h-Convex Interval-Valued Functions by Means of Pseudo Order Relation","volume":"15","author":"Khan","year":"2021","journal-title":"Appl. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.2991\/ijcis.d.210620.001","article-title":"Harmonically Convex Fuzzy-Interval-Valued Functions and Fuzzy-Interval Riemann\u2013Liouville Fractional Integral Inequalities","volume":"14","author":"Sana","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1856","DOI":"10.2991\/ijcis.d.210616.001","article-title":"Higher-Order Strongly Preinvex Fuzzy Mappings and Fuzzy Mixed Variational-Like Inequalities","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_54","first-page":"17","article-title":"On Some Characterization of Preinvex Fuzzy Mappings","volume":"5","author":"Khan","year":"2021","journal-title":"Earth. J. Math. Sci."},{"key":"ref_55","first-page":"11","article-title":"On Fuzzy Quasi-Invex Sets","volume":"9","author":"Khan","year":"2021","journal-title":"Int. J. Algeb. Stat."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"511","DOI":"10.7153\/jmi-2021-15-38","article-title":"New generalized Riemann-Liouville fractional integral inequalities for convex functions","volume":"15","author":"Mohammed","year":"2021","journal-title":"J. Math. Inequal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/175\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:38:32Z","timestamp":1760164712000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/3\/175"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,31]]},"references-count":56,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["axioms10030175"],"URL":"https:\/\/doi.org\/10.3390\/axioms10030175","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,31]]}}}