{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T15:37:07Z","timestamp":1776267427454,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,20]],"date-time":"2020-06-20T00:00:00Z","timestamp":1592611200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The authors have reviewed a wide production of scientific articles dealing with the evolution of the concept of convexity and its various applications, and based on this they have detected the relationship that can be established between trapezoidal inequalities, generalized convex functions, and special functions, in particular with the so-called Raina function, which generalizes other better known ones such as the hypergeometric function and the Mittag\u2013Leffler function. The authors approach this situation by studying the Hermite\u2013Hadamard inequality, establishing a useful identity using Raina\u2019s fractional integral operator in the setting of    \u03d5   -convex functions, obtaining some integral inequalities connected with the right-hand side of Hermite\u2013Hadamard-type inequalities for Raina\u2019s fractional integrals. Various special cases have been identified.<\/jats:p>","DOI":"10.3390\/sym12061034","type":"journal-article","created":{"date-parts":[[2020,6,24]],"date-time":"2020-06-24T07:14:19Z","timestamp":1592982859000},"page":"1034","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Trapezium-Type Inequalities for Raina\u2019s Fractional Integrals Operator Using Generalized Convex Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"first","affiliation":[{"name":"Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Cat\u00f3lica del Ecuador, Escuela de Matem\u00e1ticas y F\u00edsicas, Quito, Ecuador"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0115-3079","authenticated-orcid":false,"given":"Artion","family":"Kashuri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora, Albania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4406-5469","authenticated-orcid":false,"given":"Jorge E. Hern\u00e1ndez","family":"Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Decanato de Ciencias Econ\u00f3micas y Empresariales, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/rapstu\/raq001","article-title":"Jensen\u2019s Inequality, parameter uncertainty and multiperiod investment","volume":"1","author":"Grinalatt","year":"2011","journal-title":"Rev. Asset Pric. Stud."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Nicolescu, C., and Peerson, L. (2006). Convex Functions and Their Applications. A Contemporary Approach, Springer. CMS Books in Mathematics.","DOI":"10.1007\/0-387-31077-0_2"},{"key":"ref_3","unstructured":"Pe\u010dari\u0107, J.E., Proschan, F., and Tong, Y.L. (1992). Convex functions, partial orderings and statistical applications. Mathematics. Science and Engineering, Academic Press, Inc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/S0169-5347(99)01664-X","article-title":"Jensen\u2019s inequality predicts effects of environmental variations","volume":"14","author":"Ruel","year":"1999","journal-title":"Trends Ecol. Evol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.15517\/rmta.v26i1.36214","article-title":"Hermite-Hadamard Inequalities type for Raina\u2019s Fractional integral Operator using \u03b7-Convex Functions","volume":"26","year":"2019","journal-title":"Revista Matem\u00e1tica Teor\u00eda y Aplicaciones (Univ. Nac. Costa Rica)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"766","DOI":"10.22436\/jnsa.009.03.05","article-title":"Hermite-Hadamard type inequalities for MT-convex functions via classical integrals and fractional integrals","volume":"9","author":"Liu","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_7","first-page":"165","article-title":"Some new classes of non-convex functions","volume":"11","author":"Noor","year":"2006","journal-title":"Nonlinear Funct. Analy. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.18576\/amis\/100507","article-title":"F\u00e9jer type inequalities for (s,m)-convex functions in second sense","volume":"10","year":"2016","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"145","DOI":"10.15446\/recolma.v50n2.62207","article-title":"New Hermite-Hadamard and Jensen type inequalities for h-convex functions on fractal sets","volume":"50","author":"Vivas","year":"2016","journal-title":"Rev. Colomb. Matem\u00e1ticas"},{"key":"ref_10","first-page":"1","article-title":"Some k-fractional extensions of the trapezium inequalities through generalized relative semi-(m,h)-preinvexity","volume":"2019","author":"Du","year":"2019","journal-title":"Appl. Anal."},{"key":"ref_11","first-page":"41","article-title":"Some fractional integral inequalities of Hermite Hadamard and Minkowski type. Selecciones","volume":"6","year":"2019","journal-title":"Matem\u00e1ticas (Univ. Trujillo Per\u00fa)"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1252","DOI":"10.22436\/jnsa.009.03.50","article-title":"On Hermite-Hadamard type inequalities via fractional integral of a function with respect to another function","volume":"9","author":"Jleli","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_13","first-page":"38","article-title":"Inequalities of Fej\u00e9r type related to generalized convex functions with applications","volume":"16","author":"Aslani","year":"2018","journal-title":"Int. J. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"705","DOI":"10.22436\/jnsa.009.02.32","article-title":"Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions","volume":"9","author":"Chen","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_15","first-page":"1","article-title":"Some generalizations of Hermite-Hadamard type inequalities","volume":"5","author":"Delavar","year":"2016","journal-title":"Springerplus"},{"key":"ref_16","first-page":"103","article-title":"Some new Hermite-Hadamard type inequalities and their applications","volume":"56","author":"Kashuri","year":"2019","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_17","first-page":"1","article-title":"Some new Hermite-Hadamard integral inequalities for convex functions","volume":"1","author":"Omotoyinbo","year":"2014","journal-title":"Int. J. Sci. Innov. Tech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2012\/980438","article-title":"Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means","volume":"2012","author":"Xi","year":"2012","journal-title":"J. Funct. Spaces Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1051\/mmnp\/2018067","article-title":"Response functions in linear viscoelastic constitutive equations and related fractional operators","volume":"14","author":"Hristov","year":"2019","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.physa.2017.10.002","article-title":"Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel","volume":"492","author":"Kumar","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/i2018-11863-9","article-title":"Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative","volume":"133","author":"Owolabi","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"411","DOI":"10.5556\/j.tkjm.44.2013.1218","article-title":"Some Hermite-Hadamard type inequalities via Riemann-Liouville fractional calculus","volume":"44","author":"Mihai","year":"2013","journal-title":"Tamkang J. Math."},{"key":"ref_23","first-page":"89","article-title":"k-Fractional integrals and applications","volume":"7","author":"Mubeen","year":"2012","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_24","first-page":"153","article-title":"The Hadamard\u2019s inequality for convex function via fractional integrals","volume":"33","author":"Dragomir","year":"2013","journal-title":"Acta Math. Sci. Ser. B Engl. Ed."},{"key":"ref_25","first-page":"1","article-title":"Generalized Hermite-Hadamard type inequalities involving fractional integral operators","volume":"169","author":"Set","year":"2017","journal-title":"J. Inequalities Appl."},{"key":"ref_26","first-page":"1","article-title":"k-fractional integral trapezium-like inequalities through (h,m)-convex and (\u03b1,m)-convex mappings","volume":"311","author":"Wang","year":"2017","journal-title":"J. Inequalities Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2045","DOI":"10.18576\/amis\/100606","article-title":"Hermite-Hadamard-Fej\u00e9r type inequalities for strongly (s, m)-convex functions with modulus c, in second sense","volume":"10","author":"Bracamonte","year":"2016","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"53","DOI":"10.18576\/amisl\/060201","article-title":"Hermite-Hadamard type inequalities for harmonically convex functions on n-coordinates","volume":"6","author":"Viloria","year":"2018","journal-title":"Appl. Math. Inf. Sci. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"79","DOI":"10.18576\/amis\/110110","article-title":"Ostrowski Type inequalities for functions whose derivatives are (m,h1,h2)-convex","volume":"11","year":"2017","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"39","DOI":"10.18576\/amisl\/040201","article-title":"Relative strongly h-convex functions and integral inequalities","volume":"4","year":"2016","journal-title":"Appl. Math. Inf. Sci. Lett."},{"key":"ref_31","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_32","unstructured":"Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1955). Higher Transcendental Functions, McGraw-Hil."},{"key":"ref_33","unstructured":"Lebedev, A. (1972). Special Functions and Their Applications, Dover Publications, Inc."},{"key":"ref_34","first-page":"5","article-title":"On Ostrowski Type Inequalities","volume":"56","author":"Agarwal","year":"2016","journal-title":"Fasc. Math."},{"key":"ref_35","first-page":"48","article-title":"On weighted Montogomery identities for Riemann-Liouville fractional integrals","volume":"1","author":"Sarikaya","year":"2013","journal-title":"Konuralp J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/1034\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:41:07Z","timestamp":1760175667000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/1034"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,20]]},"references-count":35,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12061034"],"URL":"https:\/\/doi.org\/10.3390\/sym12061034","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,20]]}}}