{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T01:59:37Z","timestamp":1767923977854,"version":"3.49.0"},"reference-count":37,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T00:00:00Z","timestamp":1671408000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Spanish Government and the European Commission","award":["RTI2018-094336-B-I00"],"award-info":[{"award-number":["RTI2018-094336-B-I00"]}]},{"name":"Spanish Government and the European Commission","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}]},{"name":"Spanish Government and the European Commission","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]},{"name":"Basque Government","award":["RTI2018-094336-B-I00"],"award-info":[{"award-number":["RTI2018-094336-B-I00"]}]},{"name":"Basque Government","award":["IT1207-19"],"award-info":[{"award-number":["IT1207-19"]}]},{"name":"Basque Government","award":["IT1555-22"],"award-info":[{"award-number":["IT1555-22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we provide constraints for the sum by employing a generalized modified form of fractional integrals of Riemann-type via convex functions. The mean fractional inequalities for functions with convex absolute value derivatives are discovered. Hermite\u2013Hadamard-type fractional inequalities for a symmetric convex function are explored. These results are achieved using a fresh and innovative methodology for the modified form of generalized fractional integrals. Some applications for the results explored in the paper are briefly reviewed.<\/jats:p>","DOI":"10.3390\/sym14122682","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T06:58:29Z","timestamp":1671433109000},"page":"2682","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Study on the Modified Form of Riemann-Type Fractional Inequalities via Convex Functions and Related Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8480-2817","authenticated-orcid":false,"given":"Muhammad","family":"Samraiz","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maria","family":"Malik","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kanwal","family":"Saeed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saima","family":"Naheed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1574-1800","authenticated-orcid":false,"given":"Sina","family":"Etemad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 3751-71379, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV\/EHU), 48940 Leioa, Bizkaia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3463-2607","authenticated-orcid":false,"given":"Shahram","family":"Rezapour","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 3751-71379, Iran"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8367","DOI":"10.3934\/math.2021485","article-title":"Modeling fractional-order dynamics of Syphilis via Mittag-Leffler law","volume":"6","author":"Bonyah","year":"2021","journal-title":"AIMS Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1186\/s13661-020-01361-0","article-title":"A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions","volume":"2020","author":"Baleanu","year":"2020","journal-title":"Bound. Value Probl."},{"key":"ref_3","first-page":"105649","article-title":"Existence of results and computational analysis of a fractional order two strain epidemic model","volume":"39","author":"Khan","year":"2022","journal-title":"Res. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"110668","DOI":"10.1016\/j.chaos.2021.110668","article-title":"A theoretical study of the Caputo\u2013Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control","volume":"144","author":"Mohammadi","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"111821","DOI":"10.1016\/j.chaos.2022.111821","article-title":"Non-fractional and fractional mathematical analysis and simulations for Q fever","volume":"156","author":"Asamoah","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","first-page":"7286460","article-title":"Study of the fractional-order HIV-1 infection model with uncertainty in initial data","volume":"2022","author":"Wu","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"112511","DOI":"10.1016\/j.chaos.2022.112511","article-title":"Some novel mathematical analysis on the fractal\u2013fractional model of the AH1N1\/09 virus and its generalized Caputo-type version","volume":"162","author":"Etemad","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Najafi, H., Etemad, S., Patanarapeelert, N., Asamoah, J.K.K., Rezapour, S., and Sitthiwirattham, T. (2022). A study on dynamics of CD4+ T-cells under the effect of HIV-1 infection based on a mathematical fractal-fractional model via the Adams-Bashforth scheme and Newton polynomials. Mathematics, 10.","DOI":"10.3390\/math10091366"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Abbas, M.I., and Ragusa, M.A. (2021). On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry, 13.","DOI":"10.3390\/sym13020264"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"8414","DOI":"10.1002\/mma.6188","article-title":"Hermite-Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels","volume":"44","author":"Fernandez","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6325","DOI":"10.3934\/math.2020407","article-title":"Fractional generalized Hadamard and Fejer-Hadamard inequalities for m-convex functions","volume":"5","author":"Yang","year":"2020","journal-title":"AIMS Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Sarikaya, M.Z., and Baleanu, D. (2020). On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals. Symmetry, 12.","DOI":"10.3390\/sym12040595"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"9945114","DOI":"10.1155\/2021\/9945114","article-title":"Hadamard and Fejer-Hadamard inequalities for (\u03b1, h \u2212 m) \u2212 p-convex functions via Riemann-Liouville fractional integrals","volume":"2021","author":"Jia","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"6377","DOI":"10.3934\/math.2021374","article-title":"Fractional integral inequalities for h-convex functions via Caputo-Fabrizio operator","volume":"6","author":"Chen","year":"2021","journal-title":"AIMS Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"El Shahed, M. (2003). Fractional Calculus Model of Semilunar Heart Valve Vibrations, International Mathematica Symposium.","DOI":"10.1142\/9781848161313_0008"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1115\/1.1478062","article-title":"Application of fractional calculus to fluid mechanics","volume":"124","author":"Kulish","year":"2002","journal-title":"J. Fluids Eng."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific Publishing Company.","DOI":"10.1142\/9789812817747"},{"key":"ref_18","first-page":"6663971","article-title":"Some inequalities for a new class of convex functions with applications via local fractional integral","volume":"2021","author":"Hu","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_19","first-page":"8063803","article-title":"Ostrowski type inequalities for s-convex functions via q-integrals","volume":"2022","author":"Khan","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1186\/s13662-020-02720-y","article-title":"Generation of new fractional inequalities via n-polynomials s-type convexity with applications","volume":"2020","author":"Rashid","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1186\/s13662-020-02865-w","article-title":"Some new local fractional inequalities associated with generalized (s,m)-convex functions and applications","volume":"2020","author":"Abdeljawad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kashuri, A., Agarwal, R.P., Mohammed, P.O., Nonlaopon, K., Abualnaja, K.M., and Hamed, Y.S. (2022). New generalized class of convex functions and some related integral inequalities. Symmetry, 14.","DOI":"10.3390\/sym14040722"},{"key":"ref_23","first-page":"8575563","article-title":"Schur, Hermite-Hadamard, and Fejer type inequalities for the class of higher-order generalized convex functions","volume":"2022","author":"Ma","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Abdeljawad, T., and Kashuri, A. (2020). Fractional Hermite-Hadamard-Fejer inequalities for a convex function with respect to an increasing function involving a positive weighted symmetric function. Symmetry, 12.","DOI":"10.3390\/sym12091503"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5869","DOI":"10.22436\/jnsa.010.11.24","article-title":"Generalized harmonically convex functions on fractal sets and related Hermite-Hadamard type inequalities","volume":"10","author":"Sun","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6716830","DOI":"10.1155\/2022\/6716830","article-title":"The Hermite-Hadamard-Mercer type inequalities via generalized proportional fractional integral concerning another function","volume":"2022","author":"Aljaaidi","year":"2022","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1186\/s13660-022-02772-6","article-title":"On some Volterra-Fredholm and Hermite-Hadamard-type fractional integral inequalities","volume":"2022","author":"Bounoua","year":"2022","journal-title":"J. Inequal. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF02418571","article-title":"Sur les fonctions convexes et les inegalites entre les valeurs moyennes","volume":"30","author":"Jensen","year":"1906","journal-title":"Acta Math."},{"key":"ref_29","first-page":"79","article-title":"On Hadamard inequalities for k-fractional integrals","volume":"4","author":"Farid","year":"2016","journal-title":"Konuralp J. Math."},{"key":"ref_30","first-page":"179","article-title":"On hypergeometric functions and Pochhammer k-symbol","volume":"15","author":"Diaz","year":"2007","journal-title":"Divulg. Mat."},{"key":"ref_31","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier. [1st ed.]. North-Holland Mathematics Studies: 204."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"64946","DOI":"10.1109\/ACCESS.2018.2878266","article-title":"Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities","volume":"6","author":"Kwun","year":"2018","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Wu, S., Samraiz, M., Mehmood, A., Jarad, F., and Naheed, S. (2022). Some symmetric properties and applications of weighted fractional integral operator. Symmetry, submitted.","DOI":"10.1142\/S0218348X2340011X"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"37","DOI":"10.31801\/cfsuasmas.484437","article-title":"Study of a generalized Riemann-Liouville fractional integral via convex functions","volume":"69","author":"Farid","year":"2020","journal-title":"Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Farid, G., Nazeer, W., Saleem, M.S., Mehmood, S., and Kang, S.M. (2018). Bounds of Riemann-Liouville fractional integrals in general form via convex functions and their applications. Mathematics, 6.","DOI":"10.3390\/math6110248"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1007\/s41478-018-0079-4","article-title":"Some Riemann-Liouville fractional integral inequalities for convex functions","volume":"27","author":"Farid","year":"2019","journal-title":"J. Anal."},{"key":"ref_37","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."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2682\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:48Z","timestamp":1760147028000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2682"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,19]]},"references-count":37,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["sym14122682"],"URL":"https:\/\/doi.org\/10.3390\/sym14122682","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,19]]}}}