{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T19:18:38Z","timestamp":1773515918387,"version":"3.50.1"},"reference-count":114,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,19]],"date-time":"2022-04-19T00:00:00Z","timestamp":1650326400000},"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>Inequalities play important roles not only in mathematics but also in other fields, such as economics and engineering. Even though many results are published as Hermite\u2013Hadamard (H-H)-type inequalities, new researchers to these fields often find it difficult to understand them. Thus, some important discoverers, such as the formulations of H-H-type inequalities of \u03b1-type real-valued convex functions, along with various classes of convexity through differentiable mappings and for fractional integrals, are presented. Some well-known examples from the previous literature are used as illustrations. In the many above-mentioned inequalities, the symmetrical behavior arises spontaneously.<\/jats:p>","DOI":"10.3390\/sym14050840","type":"journal-article","created":{"date-parts":[[2022,4,20]],"date-time":"2022-04-20T00:22:43Z","timestamp":1650414163000},"page":"840","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Review of Hermite\u2013Hadamard Inequality for \u03b1-Type Real-Valued Convex Functions"],"prefix":"10.3390","volume":"14","author":[{"given":"Ohud","family":"Almutairi","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Hafr Al Batin, Hafr Al-Batin 31991, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1217-963X","authenticated-orcid":false,"given":"Adem","family":"K\u0131l\u0131\u00e7man","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Universiti Putra Malaysia (UPM), Serdang 43400, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/BF02189414","article-title":"Hermite and convexity","volume":"28","year":"1985","journal-title":"Aequ. Math."},{"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":"58","author":"Hadamard","year":"1893","journal-title":"J. Math. Pures Appl."},{"key":"ref_3","first-page":"1","article-title":"Sur deux limites d\u2019une int\u00e9grale d\u00e9finie","volume":"3","author":"Hermite","year":"1883","journal-title":"Mathesis"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dragomir, S., and Pearce, C. (2004). Selected Topics on Hermite\u2013Hadamard Inequalities and Applications, Austral Internet Publishing. RGMIA Monographs.","DOI":"10.1023\/B:APOM.0000027220.51557.6d"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bullen, P. (2003). Handbook of Means and Their Inequalities, Kluwer Academic Publisher.","DOI":"10.1007\/978-94-017-0399-4"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wang, J., and Fe\u010dkan, M. (2018). Fractional Hermite-Hadamard Inequalities, Walter de Gruyter.","DOI":"10.1515\/9783110523621"},{"key":"ref_7","unstructured":"Almutairi, O. (2020). Generalization of Hermite-Hadamard type inequalities and their applications. [Ph.D. Thesis, Universiti Putra Malaysia]."},{"key":"ref_8","first-page":"1","article-title":"New fractional inequalities of midpoint type via s-convexity and their application","volume":"2019","author":"Almutairi","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"20479","DOI":"10.1109\/ACCESS.2019.2897680","article-title":"New quantum Hermite-Hadamard inequalities utilizing harmonic convexity of the functions","volume":"7","author":"Awan","year":"2019","journal-title":"IEEE Access"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"124059","DOI":"10.1016\/j.jmaa.2020.124059","article-title":"Convexity according to a pair of quasi-arithmetic means and inequalities","volume":"488","author":"Duc","year":"2020","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0893-9659(98)00086-X","article-title":"Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula","volume":"11","author":"Dragomir","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_12","first-page":"26","article-title":"Hadamard-type inequalities for s-convex functions","volume":"193","author":"Kirmaci","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.insmatheco.2019.10.007","article-title":"Convex risk functionals: Representation and applications","volume":"90","author":"Liu","year":"2020","journal-title":"Insur. Math. Econ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"568","DOI":"10.3390\/sym12040568","article-title":"New generalized Hermite-Hadamard inequality and related integral inequalities involving Katugampola type fractional integrals","volume":"12","author":"Almutairi","year":"2020","journal-title":"Symmetry"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2364","DOI":"10.1002\/mma.5893","article-title":"Hermite-Hadamard type inequalities for generalized Riemann-Liouville fractional integrals of h-convex functions","volume":"44","author":"Dragomir","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Dahmani, Z., and Belhamiti, M.M. (2020). Integral Inequalities and Differential Equations via Fractional Calculus, IntechOpen.","DOI":"10.5772\/intechopen.91140"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"110938","DOI":"10.1016\/j.chaos.2021.110938","article-title":"Generalized Fej\u00e9r\u2013Hermite\u2013Hadamard type via generalized (h- m)-convexity on fractal sets and applications","volume":"147","author":"Almutairi","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","unstructured":"Udriste, C. (2013). Convex Functions and Optimization Methods on Riemannian Manifolds, Springer."},{"key":"ref_19","first-page":"1","article-title":"A note on generalized convex functions","volume":"2019","author":"Ullah","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_20","first-page":"49","article-title":"Om konvekse funktioner og uligheder imellem middelvaerdier","volume":"16","author":"Jensen","year":"1905","journal-title":"Nyt Tidsskr. Mat."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Niculescu, C., and Persson, L.E. (2006). Convex Functions and Their Applications, Springer.","DOI":"10.1007\/0-387-31077-0"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mo, H., Sui, X., and Yu, D. (2014). Generalized convex functions and some inequalities on fractal sets. arXiv.","DOI":"10.1155\/2014\/636751"},{"key":"ref_23","first-page":"138","article-title":"Neravenstva dlja funkcii sirokogo klassa, soderzascego vypuklye, monotonnye i nekotorye drugie vidy funkii","volume":"166","author":"Godunova","year":"1985","journal-title":"Vycislitel. Mat. i Fiz. Mezvuzov. Sb. Nauc. Tr. Mgpi Mosk."},{"key":"ref_24","unstructured":"Mitrinovi\u0107, D.S., Pe\u010dari\u0107, J., and Fink, A.M. (2013). Classical and New Inequalities in Analysis, Springer Science and Business Media."},{"key":"ref_25","first-page":"335","article-title":"Some inequalities of Hadamard type","volume":"21","author":"Dragomir","year":"1995","journal-title":"Soochow J. Math"},{"key":"ref_26","first-page":"853","article-title":"On the Godunova-Levin-Schur class of functions","volume":"12","author":"Radulescu","year":"2009","journal-title":"Math. Inequal. Appl"},{"key":"ref_27","first-page":"1","article-title":"On the (p, h)-convex function and some integral inequalities","volume":"2014","author":"Fang","year":"2014","journal-title":"J. Inequalities Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.17776\/csj.358766","article-title":"Some new integral inequalities for n-times differentiable Godunova-Levin functions","volume":"38","author":"Kadakal","year":"2017","journal-title":"Cumhur. Sci. J."},{"key":"ref_29","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_30","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":"Aequ. Math."},{"key":"ref_31","first-page":"687","article-title":"The Hadamard inequalities for s-convex functions in the second sense","volume":"32","author":"Dragomir","year":"1999","journal-title":"Demonstr. Math."},{"key":"ref_32","first-page":"358","article-title":"A generalization of Simpson\u2019s inequality via differentiable mapping using extended (s, m)-convex functions","volume":"293","author":"Du","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.2298\/FIL1806153U","article-title":"On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators","volume":"32","author":"Usta","year":"2018","journal-title":"Filomat"},{"key":"ref_34","first-page":"87","article-title":"Some generalized Hermite-Hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex","volume":"88","author":"Gozpinar","year":"2019","journal-title":"Acta Math. Univ. Comen."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s00010-007-2891-9","article-title":"Exploring the concept of s-convexity","volume":"74","author":"Pinheiro","year":"2007","journal-title":"Aequ. Math."},{"key":"ref_36","first-page":"43","article-title":"The Jensen inequality for s-Breckner convex functions in linear spaces","volume":"33","author":"Dragomir","year":"2000","journal-title":"Demonstr. Math."},{"key":"ref_37","first-page":"629","article-title":"The Hadamard\u2019s inequality for s-convex function of 2-variables on the co-ordinates","volume":"2","author":"Alomari","year":"2008","journal-title":"Int. J. Math. Anal."},{"key":"ref_38","first-page":"45","article-title":"Integral inequalities of Jensen type for \u03bb-convex functions","volume":"68","author":"Dragomir","year":"2016","journal-title":"Mat. Vesn."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1186\/s13660-015-0826-x","article-title":"Notions of generalized s-convex functions on fractal sets","volume":"2015","author":"Saleh","year":"2015","journal-title":"J. Inequalities Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"11","DOI":"10.17512\/jamcm.2016.4.02","article-title":"Generalized Ostrowski type inequalities for functions whose local fractional derivatives are generalized s-convex in the second sense","volume":"15","author":"Budak","year":"2016","journal-title":"J. Appl. Math. Comput. Mech."},{"key":"ref_41","first-page":"167","article-title":"Fractional Ostrowski inequalities for s-Godunova-Levin functions","volume":"5","author":"Noor","year":"2014","journal-title":"Int. J. Anal. Appl."},{"key":"ref_42","first-page":"63","article-title":"Hermite-Hadamard type inequalities for generalized (s,m)-preinvex Godunova-Levin functions","volume":"534","author":"Kashuri","year":"2018","journal-title":"Mat. Znan."},{"key":"ref_43","first-page":"1","article-title":"What is invexity?","volume":"28","author":"Mond","year":"1986","journal-title":"Anziam J."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/0022-247X(81)90123-2","article-title":"On sufficiency of the Kuhn-Tucker conditions","volume":"80","author":"Hanson","year":"1981","journal-title":"J. Math. Anal. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0022-247X(88)90113-8","article-title":"Pre-invex functions in multiple objective optimization","volume":"136","author":"Weir","year":"1988","journal-title":"J. Math. Anal. Appl."},{"key":"ref_46","first-page":"5","article-title":"On Hadamard-type inequalities for s-preinvex functions","volume":"27","author":"Li","year":"2010","journal-title":"J. Chongqing Norm. Univ."},{"key":"ref_47","first-page":"170","article-title":"Hadamard type inequalities for (s, r) preinvex functions in the first sense","volume":"5","author":"Meftah","year":"2017","journal-title":"Electron. J. Math. Anal. Appl."},{"key":"ref_48","first-page":"73","article-title":"Fractional Hermite-Hadamard type inequalities for functions whose derivatives are extended s-(\u03b1, m)-preinvex","volume":"9","author":"Meftah","year":"2019","journal-title":"Int. J. Optim. Control: Theor. Appl. (IJOCTA)"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1109\/MIE.2007.901479","article-title":"Fractional calculus: A mathematical tool from the past for present engineers [Past and present]","volume":"1","author":"Cafagna","year":"2007","journal-title":"IEEE Ind. Electron. Mag."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1140","DOI":"10.1016\/j.cnsns.2010.05.027","article-title":"Recent history of fractional calculus","volume":"16","author":"Machado","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Yang, X.J. (2019). General Fractional Derivatives: Theory, Methods and Applications, CRC Press.","DOI":"10.1201\/9780429284083"},{"key":"ref_52","first-page":"47","article-title":"Mathematical and physical interpretations of fractional derivatives and integrals","volume":"1","author":"Hilfer","year":"2019","journal-title":"Handb. Fract. Calc. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Machado, J.A.T., and Luo, A.C. (2011). Fractional Dynamics and Control, Springer Science and Business Media.","DOI":"10.1007\/978-1-4614-0457-6"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1007\/BF01036529","article-title":"Fractional integral and its physical interpretation","volume":"90","author":"Nigmatullin","year":"1992","journal-title":"Theor. Math. Phys."},{"key":"ref_55","unstructured":"Kilbas, A.A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science and Business Media.","DOI":"10.1007\/978-3-642-14574-2"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2011). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science and Business Media.","DOI":"10.1007\/978-3-642-14003-7"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Malinowska, A.B., Odzijewicz, T., and Torres, D.F. (2015). Advanced Methods in the Fractional Calculus of Variations, Springer.","DOI":"10.1007\/978-3-319-14756-7"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.cnsns.2018.05.011","article-title":"Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems","volume":"67","author":"Salati","year":"2019","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_60","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives, Gordon and Breach Science Publishers."},{"key":"ref_61","first-page":"1191","article-title":"Hadamard-type fractional calculus","volume":"38","author":"Anatoly","year":"2001","journal-title":"J. Korean Math. Soc."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/S0022-247X(02)00049-5","article-title":"Compositions of Hadamard-type fractional integration operators and the semigroup property","volume":"269","author":"Butzer","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0022-247X(02)00001-X","article-title":"Fractional calculus in the Mellin setting and Hadamard-type fractional integrals","volume":"269","author":"Butzer","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_64","first-page":"566","article-title":"Mellin transforms of generalized fractional integrals and derivatives","volume":"257","author":"Katugampola","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_65","first-page":"860","article-title":"New approach to a generalized fractional integral","volume":"218","author":"Katugampola","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1134\/S0005117913040012","article-title":"Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation","volume":"74","author":"Butkovskii","year":"2013","journal-title":"Autom. Remote Control"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1186\/1029-242X-2013-167","article-title":"Some relations involving a generalized fractional derivative operator","volume":"2013","author":"Gaboury","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_68","unstructured":"Richard, H. (2014). Fractional Calculus: An Introduction for Physicists, World Scientific."},{"key":"ref_69","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_70","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2014.01.002","article-title":"A new definition of fractional derivative","volume":"264","author":"Khalil","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_71","first-page":"6926107","article-title":"Conformable integral inequalities of the Hermite-Hadamard type in terms of GG-and GA-convexities","volume":"2019","author":"Khurshid","year":"2019","journal-title":"J. Funct. Spaces"},{"key":"ref_72","first-page":"9845407","article-title":"Some new Hermite\u2013Hadamard-type inequalities associated with conformable fractional integrals and their applications","volume":"2020","author":"Iqbal","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_73","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D.S., and Vasic, P.M. (1970). Analytic Inequalities, Springer.","DOI":"10.1007\/978-3-642-99970-3"},{"key":"ref_74","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":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"663","DOI":"10.14321\/realanalexch.29.2.0663","article-title":"Old and new on the Hermite-Hadamard inequality","volume":"29","author":"Niculescu","year":"2004","journal-title":"Real Anal. Exch."},{"key":"ref_76","first-page":"273","article-title":"Three proofs of the inequality","volume":"117","author":"Khattri","year":"2010","journal-title":"Am. Math. Mon."},{"key":"ref_77","unstructured":"Hardy, G., Littlewood, J., and Polya, G. (1952). Inequalities, Cambrige University Press."},{"key":"ref_78","unstructured":"Robert, A.W., and Varberg, D.E. (1973). Convex Functions, Academic Press."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1006\/jmaa.1999.6593","article-title":"P-functions, quasi-convex functions, and Hadamard-type inequalities","volume":"240","author":"Pearce","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1017\/S0004972711003029","article-title":"Hermite\u2013Hadamard type inequalities for functions when a power of the absolute value of the first derivative is P-convex","volume":"86","author":"Barani","year":"2012","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"78","DOI":"10.11121\/ijocta.01.2020.00787","article-title":"Some Hermite-Hadamard type inequalities for (P; m)-function and quasi m-convex functions","volume":"10","author":"Kadakal","year":"2020","journal-title":"Int. J. Optim. Control Theor. Appl."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1186\/1029-242X-2012-21","article-title":"On some Hadamard-type inequalities for product of two s-convex functions on the co-ordinates","volume":"2012","author":"Latif","year":"2012","journal-title":"J. Inequalities Appl."},{"key":"ref_83","first-page":"306","article-title":"On new general integral inequalities for s-convex functions","volume":"246","author":"Set","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_84","first-page":"17","article-title":"A mapping in connection to Hadamard\u2019s inequalities","volume":"128","author":"Dragomir","year":"1991","journal-title":"Akad. Der Wissenschaften. Math. Nat. Kl."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/0022-247X(92)90233-4","article-title":"Two mappings in connection to Hadamard\u2019s inequalities","volume":"167","author":"Dragomir","year":"1992","journal-title":"J. Math. Anal. Appl."},{"key":"ref_86","first-page":"13","article-title":"Some integral inequalities for differentiable convex functions","volume":"13","author":"Dragomir","year":"1992","journal-title":"Contrib. Maced. Acad. Sci. Arts"},{"key":"ref_87","unstructured":"Dragomir, S.S., Milo\u0161evi\u0107, D.M., and S\u00e1ndor, J. (1993). On some refinements of Hadamard\u2019s inequalities and applications. Publikacije Elektrotehni\u010dkog Fakulteta. Serija Matematika, University of Belgrade."},{"key":"ref_88","first-page":"357","article-title":"New refinements of the Hermite-Hadamard integral inequality for convex functions and applications","volume":"28","author":"Dragomir","year":"2002","journal-title":"Soochow J. Math."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"45","DOI":"10.5556\/j.tkjm.34.2003.271","article-title":"Further properties of some mappings associated with Hermite-Hadamard inequalities","volume":"34","author":"Dragomir","year":"2003","journal-title":"Tamkang J. Math."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1186\/s13662-015-0639-8","article-title":"Some generalized Hermite-Hadamard type integral inequalities for generalized s-convex functions on fractal sets","volume":"2015","author":"Saleh","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_91","first-page":"126","article-title":"Hermite-Hadamard integral inequalities for log-preinvex functions","volume":"2","author":"Noor","year":"2007","journal-title":"J. Math. Anal. Approx. Theory"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0893-9659(99)00164-0","article-title":"Inequalities for differentiable mappings with application to special means and quadrature formulae","volume":"13","author":"Pearce","year":"2000","journal-title":"Appl. Math. Lett."},{"key":"ref_93","first-page":"137","article-title":"Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula","volume":"147","author":"Kirmaci","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1186\/1029-242X-2012-247","article-title":"Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex","volume":"2012","author":"Barani","year":"2012","journal-title":"J. Inequalities Appl."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/j.cam.2018.10.022","article-title":"New Hermite-Hadamard type integral inequalities for convex functions and their applications","volume":"350","author":"Mehrez","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"21","DOI":"10.2478\/v10294-012-0011-5","article-title":"Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula","volume":"8","author":"Zhu","year":"2012","journal-title":"J. Appl. Math. Stat. Inform."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"2241","DOI":"10.1080\/00036811.2012.727986","article-title":"Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity","volume":"92","author":"Wang","year":"2013","journal-title":"Appl. Anal."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"69","DOI":"10.2478\/jamsi-2014-0014","article-title":"The Hermite-Hadamard\u2019s inequality for some convex functions via fractional integrals and related results","volume":"10","author":"Set","year":"2014","journal-title":"J. Appl. Math. Stat. Inform."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"655","DOI":"10.36045\/bbms\/1382448186","article-title":"Refinements of Hermite-Hadamard type inequalities involving fractional integrals","volume":"20","author":"Wang","year":"2013","journal-title":"Bull. Belg. Math. Soc. -Simon Stevin"},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s40096-017-0227-z","article-title":"Hermite-Hadamard-type inequalities for generalized s-convex functions on real linear fractal set R\u03b1(0<\u03b1<1)","volume":"11","author":"Mo","year":"2017","journal-title":"Math. Sci."},{"key":"ref_101","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. De Matem\u00c1ticas"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"109547","DOI":"10.1016\/j.chaos.2019.109547","article-title":"Fej\u00e9r\u2013Hermite\u2013Hadamard type inequalities involving generalized h-convexity on fractal sets and their applications","volume":"131","author":"Luo","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"1274","DOI":"10.1016\/j.jmaa.2016.09.018","article-title":"Hermite-Hadamard and Hermite-Hadamard\u2013Fej\u00e9r type inequalities for generalized fractional integrals","volume":"446","author":"Chen","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1186\/s13660-018-1807-7","article-title":"Integral inequalities for some convex functions via generalized fractional integrals","volume":"2018","author":"Mehreen","year":"2018","journal-title":"J. Inequalities Appl."},{"key":"ref_105","doi-asserted-by":"crossref","unstructured":"Anderson, D.R. (2016). Taylor\u2019s formula and integral inequalities for conformable fractional derivatives. Contributions in Mathematics and Engineering, Springer.","DOI":"10.1007\/978-3-319-31317-7_2"},{"key":"ref_106","first-page":"605","article-title":"Hermite-Hadamard type inequalities for quasi-convex functions via Katugampola fractional integrals","volume":"16","author":"Set","year":"2018","journal-title":"Int. J. Anal. Appl."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"273","DOI":"10.30538\/oms2019.0070","article-title":"On Hermite\u2013Hadamard\u2013Fej\u00e9r type integral inequalities for generalized convex functions via local fractional integrals","volume":"3","author":"Sarikaya","year":"2019","journal-title":"Open J. Math. Sci."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"1643","DOI":"10.1016\/S0252-9602(11)60350-0","article-title":"Some inequalities of Hermite-Hadamard type for s-convex functions","volume":"31","author":"Alomari","year":"2011","journal-title":"Acta Math. Sci."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"1539","DOI":"10.1007\/s13398-017-0444-1","article-title":"Certain Hermite-Hadamard type inequalities involving generalized fractional integral operators","volume":"112","author":"Set","year":"2018","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00edsicas Nat. Serie A. Mat."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"3882","DOI":"10.1002\/mma.4270","article-title":"Some inequalities involving Hadamard-type k-fractional integral operators","volume":"40","author":"Agarwal","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"491","DOI":"10.18514\/MMN.2015.1099","article-title":"Some new inequalities of Hermite-Hadamard type for s-convex functions","volume":"16","author":"Sarikaya","year":"2015","journal-title":"Miskolc Math. Notes"},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/S0893-9659(97)00142-0","article-title":"Applications of Ostrowski\u2019s inequality to the estimation of error bounds for some special means and for some numerical quadrature rules","volume":"11","author":"Dragomir","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_113","unstructured":"Bullen, P.S., Mitrinovi\u0107, D.S., and Vasic, M. (2013). Means and Their Inequalities, Springer Science and Business Media."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"6874","DOI":"10.3934\/math.2020441","article-title":"New Hermite\u2013Hadamard type inequalities for exponentially convex functions and applications","volume":"5","author":"Zhou","year":"2020","journal-title":"AIMS Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/840\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:56:20Z","timestamp":1760136980000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/840"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,19]]},"references-count":114,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050840"],"URL":"https:\/\/doi.org\/10.3390\/sym14050840","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,19]]}}}