{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:00:10Z","timestamp":1760230810992,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,15]],"date-time":"2022-08-15T00:00:00Z","timestamp":1660521600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Fund of Khon Kaen University, Thailand"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Integral inequalities make up a comprehensive and prolific field of research within the field of mathematical interpretations. Integral inequalities in association with convexity have a strong relationship with symmetry. Different disciplines of mathematics and applied sciences have taken a new path as a result of the development of new fractional operators. Different new fractional operators have been used to improve some mathematical inequalities and to bring new ideas in recent years. To take steps forward, we prove various Gr\u00fcss-type and Chebyshev-type inequalities for integrable functions in the frame of non-conformable fractional integral operators. The key results are proven using definitions of the fractional integrals, well-known classical inequalities, and classical relations.<\/jats:p>","DOI":"10.3390\/sym14081691","type":"journal-article","created":{"date-parts":[[2022,8,15]],"date-time":"2022-08-15T23:44:03Z","timestamp":1660607043000},"page":"1691","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Some New Fractional Integral Inequalities Pertaining to Generalized Fractional Integral Operator"],"prefix":"10.3390","volume":"14","author":[{"given":"Omar Mutab","family":"Alsalami","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4524-1951","authenticated-orcid":false,"given":"Soubhagya Kumar","family":"Sahoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Institute of Technical Education and Research, Siksha \u2019O\u2019 Anusandhan University, Bhubaneswar 751030, India"},{"name":"Department of Mathematics, Aryan Institute of Engineering and Technology, Bhubaneswar 752050, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8372-2532","authenticated-orcid":false,"given":"Muhammad","family":"Tariq","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"},{"name":"Department of Mathematics, Baluchistan Residential College Loralai, Loralai 84800, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3084-922X","authenticated-orcid":false,"given":"Asif Ali","family":"Shaikh","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"El Shaed, M.A. (2003, January 10\u201313). Fractional Calculus Model of Semilunar Heart Valve Vibrations. Proceedings of the International Mathematica Symposium, London, UK.","DOI":"10.1142\/9781848161313_0008"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3187","DOI":"10.1016\/j.aej.2020.07.040","article-title":"A new fractional-order compartmental disease model","volume":"59","author":"Hoan","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3945","DOI":"10.1016\/j.aej.2021.02.057","article-title":"The dynamics of fractional order Hepatitis B virus model with asymptomatic carriers","volume":"60","author":"Gul","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Baleanu, D., G\u00fcven\u00e7, Z.B., and Machado, J.T. (2010). New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_6","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_7","unstructured":"Magin, R.L. (2006). Fractional Calculus in Bio-Engineering, Begell House Inc. Publishers."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Atangana, A. (2016). Application of fractional calculus to epidemiology. Fract. Dyn., 174\u2013190. Warsaw, Poland: De Gruyter Open Poland.","DOI":"10.1515\/9783110472097-011"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"110776","DOI":"10.1016\/j.chaos.2021.110776","article-title":"Artificial macro-economics: A chaotic discrete-time fractional-order laboratory model","volume":"145","author":"Chu","year":"2021","journal-title":"Chaos Solitons Fract."},{"key":"ref_10","unstructured":"Axtell, M., and Bise, M.E. (1990, January 21\u201325). Fractional calculus application in control systems. Proceedings of the IEEE Conference on Aerospace and Electronics, Dayton, OH, USA."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Sahoo, S.K., Tariq, M., Ahmad, H., Aly, A.A., Felemban, B.F., and Thounthong, P. (2021). Some Hermite-Hadamard-type fractional integral inequalities involving twice-differentiable mappings. Symmetry, 13.","DOI":"10.3390\/sym13112209"},{"key":"ref_12","first-page":"1","article-title":"Certain inequalities via generalized proportional Hadamard fractional integral operators","volume":"454","author":"Rahman","year":"2019","journal-title":"Adv. Diff. Eqs."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Rashid, S., Abdeljawad, T., Jarad, F., and Noor, M.N. (2019). Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications. Mathematics, 7.","DOI":"10.3390\/math7090807"},{"key":"ref_14","first-page":"1","article-title":"The Minkowski inequalities via generalized proportional fractional integral operators","volume":"287","author":"Rahman","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Sahoo, S.K., Ahmad, H., Tariq, M., Kodamasingh, B., Aydi, H., and De la Sen, M. (2021). Hermite-Hadamard type inequalities involving k-fractional operator for (h,m)-convex Functions. Symmetry, 13.","DOI":"10.3390\/sym13091686"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Saleem, N., Ishtiaq, U., Guran, L., and Bota, M.F. (2022). On Graphical Fuzzy Metric Spaces with Application to Fractional Differential Equations. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6050238"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"12718","DOI":"10.3934\/math.2021734","article-title":"Some new generalizations of F-contraction type mappings that weaken certain conditions on Caputo fractional type differential equations","volume":"6","author":"Saleem","year":"2021","journal-title":"Aims Math."},{"key":"ref_18","first-page":"215","article-title":"Uber das maximum des absoluten Betrages von 1b\u2212a\u222babf(x)g(x)dx\u22121(b\u2212a)2\u222babf(x)dx\u222babg(x)dx","volume":"39","year":"1935","journal-title":"Math. Z."},{"key":"ref_19","first-page":"1","article-title":"Integral inequalities for Riemann-Liouville fractional integrals of a function with respect to another function","volume":"13","author":"Kacar","year":"2018","journal-title":"Iran. J. Math. Sci. Inform."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Rashid, S., Noor, M.A., Noor, K.I., Safdar, F., and Chu, Y.M. (2019). Hermite-Hadamard inequalities for the class of convex functions on time scale. Mathematics, 7.","DOI":"10.3390\/math7100956"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/0031-8914(72)90081-X","article-title":"Inequality for convex functions in quantum-statistical mechanics","volume":"59","author":"Okubo","year":"1972","journal-title":"Physica"},{"key":"ref_22","first-page":"563096","article-title":"Fractional integral inequalities via Hadamard\u2019s fractional integral","volume":"11","author":"Sudsutad","year":"2014","journal-title":"Abstract. Appl. Anal."},{"key":"ref_23","first-page":"1","article-title":"History, variations and generalisations of the Cebysev inequality and the question of some priorities","volume":"461\/497","year":"1974","journal-title":"Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"869434","DOI":"10.1155\/2014\/869434","article-title":"Some new Riemann-Liouville fractional integral inequalities","volume":"2014","author":"Tariboon","year":"2014","journal-title":"Int. J. Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Balasubramanian, S. (2015). On the Gr\u00fcss inequality for unital 2-positive linear maps. arXiv.","DOI":"10.7153\/oam-10-38"},{"key":"ref_26","first-page":"159","article-title":"Some extensions of Gr\u00fcssi\u2019 inequality and its applications","volume":"13","author":"Izumino","year":"2020","journal-title":"Nihonkai Math. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1664","DOI":"10.1002\/mma.6869","article-title":"Jensen-Gr\u00fcss inequality and its applications for the Zipf-Mandelbrot law","volume":"44","author":"Butt","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_28","first-page":"402","article-title":"Gr\u00fcss type inequalities for fractional integral operator involving the extended generalized Mittag-Leffler function","volume":"19","author":"Set","year":"2020","journal-title":"Appl. Comput. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"12559","DOI":"10.1002\/mma.7563","article-title":"Gr\u00fcss type inequalities via generalized fractional operators","volume":"44","author":"Butt","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.3934\/math.2020070","article-title":"Some Gr\u00fcss-type inequalities using generalized Katugampola fractional integral","volume":"5","author":"Aljaaidi","year":"2020","journal-title":"AIMS Math."},{"key":"ref_31","first-page":"663","article-title":"A Note on Gr\u00fcss type inequalities on time scales","volume":"17","author":"Sarikaya","year":"2008","journal-title":"Dyn. Syst. Appl."},{"key":"ref_32","first-page":"29","article-title":"A note on Chebyshev-Gr\u00fcss type inequalities for diferential functions","volume":"22","author":"Pachpatte","year":"2006","journal-title":"Tamsui Oxford J. Math. Sci."},{"key":"ref_33","first-page":"1","article-title":"On Gr\u00fcss inequalities within generalized K-fractional integrals","volume":"203","author":"Rashid","year":"2020","journal-title":"Adv. Diff. Equ."},{"key":"ref_34","first-page":"92","article-title":"New generalisation of Gr\u00fcss inequality using RiemannLiouville fractional integrals","volume":"2","author":"Dahmani","year":"2012","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_35","first-page":"1","article-title":"On some new Gr\u00fcss-type inequality using Hadamard fractional integral operator","volume":"5","author":"Chinchane","year":"2014","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_36","first-page":"26","article-title":"On an inequality of Gr\u00fcss type via variant of Pompeiu\u2019s mean value theorem","volume":"2","author":"Sarikaya","year":"2014","journal-title":"Pure Appl. Math. Lett."},{"key":"ref_37","first-page":"57","article-title":"On Gr\u00fcss type inequalities for a hypergeometric fractional integral","volume":"LXVI","author":"Kalla","year":"2011","journal-title":"Le Matematiche"},{"key":"ref_38","first-page":"1","article-title":"On new Gr\u00fcss type inequalities for conformable fractional integrals","volume":"9","author":"Mumcu","year":"2019","journal-title":"TWMS J. Appl. Eng. Math."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Valdes, J.E.N., Rodriguez, J.M., and Sigarreta, J.M. (2019). New Hermite-Hadamard type inequalities involving non-conformable integral operators. Symmetry, 11.","DOI":"10.3390\/sym11091108"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1691\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:08:50Z","timestamp":1760141330000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1691"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,15]]},"references-count":39,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081691"],"URL":"https:\/\/doi.org\/10.3390\/sym14081691","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,8,15]]}}}