{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T15:54:28Z","timestamp":1768319668871,"version":"3.49.0"},"reference-count":58,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,2]],"date-time":"2023-08-02T00:00:00Z","timestamp":1690934400000},"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>This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful methods to help with the learning of key mathematical ideas. The kernel of the general family of weighted fractional integral operators is related to a wide variety of extensions and generalizations of the Mittag-Leffler function and the Hurwitz-Lerch zeta function. It delves into the applications of fractional-order integral and derivative operators in mathematical and engineering sciences. Furthermore, this article derives specific cases for selected functions and presents various applications to illustrate the obtained results. Additionally, novel applications involving the Digamma function are introduced.<\/jats:p>","DOI":"10.3390\/sym15081522","type":"journal-article","created":{"date-parts":[[2023,8,2]],"date-time":"2023-08-02T11:04:38Z","timestamp":1690974278000},"page":"1522","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Results on Minkowski-Type Inequalities for Weighted Fractional Integral Operators"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Seoul 02447, Republic of Korea"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"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, C.V. Raman Global University, Bhubaneswar 752054, India"}],"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, Iraq"}],"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 and Natural Sciences, University \u201cIsmail Qemali\u201d, 9400 Vlora, Albania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8833-6585","authenticated-orcid":false,"given":"Nejmeddine","family":"Chorfi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/3779"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"639801","DOI":"10.1155\/2010\/639801","article-title":"Some applications of fractional calculus in engineering","volume":"2010","author":"Silva","year":"2010","journal-title":"Math. Prob. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/S0378-4371(00)00255-7","article-title":"Fractional calculus and continuous-time finance","volume":"284","author":"Scalas","year":"2000","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Assaleh, K., and Ahmad, W.M. (2007, January 12\u201315). Modeling of speech signals using fractional calculus. Proceedings of the 2007 9th International Symposium on Signal Processing and Its Applications, Sharjah, United Arab Emirates.","DOI":"10.1109\/ISSPA.2007.4555563"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Atanackovic, T.M., Pilipovic, S., Stankovic, B., and Zorica, D. (2014). Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, John Wiley & Sons.","DOI":"10.1002\/9781118577530"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Atanackovic, T.M., Pilipovic, S., Stankovic, B., and Zorica, D. (2014). Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles, John Wiley & Sons.","DOI":"10.1002\/9781118909065"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1115\/1.3167616","article-title":"Applications of fractional calculus to the theory of viscoelasticity","volume":"51","author":"Koeller","year":"1984","journal-title":"J. Appl. Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4667","DOI":"10.1090\/proc\/13883","article-title":"Periodic orbit analysis for the delayed Filippov system","volume":"146","author":"Cai","year":"2018","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.nonrwa.2017.10.003","article-title":"Bifurcation of limit cycles at infinity in piecewise polynomial systems","volume":"41","author":"Chen","year":"2018","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"106163","DOI":"10.1016\/j.rinp.2022.106163","article-title":"New soliton solutions and modulation instability analysis of fractional Huxley equation","volume":"44","author":"Rahman","year":"2023","journal-title":"Results Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"136","DOI":"10.3390\/fractalfract7020136","article-title":"The sensitive visualization and generalized fractional solitons\u2019 construction for regularized long-wave governing model","volume":"7","author":"Faridi","year":"2023","journal-title":"Fractal Fract."},{"key":"ref_12","first-page":"661","article-title":"On Cauchy problems with Caputo-Hadamard fractional derivatives","volume":"40","author":"Adjabi","year":"2016","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1186\/1029-242X-2013-303","article-title":"Inequality estimates for the boundedness of multilinear singular and fractional integral operators","volume":"2013","author":"Zhou","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.1016\/j.camwa.2019.01.007","article-title":"Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch\u2013Torrey equations on irregular convex domains","volume":"78","author":"Liu","year":"2019","journal-title":"Comput. Math. Appl."},{"key":"ref_15","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science B.V.. North-Holland Mathematics Studies."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kashuri, A., Sahoo, S.K., Mohammed, P.O., Al-Sarairah, E., and Hamed, Y.S. (2023). Some New Hermite-Hadamard Type Inequalities Pertaining to Fractional Integrals with an Exponential Kernel for Subadditive Functions. Symmetry, 15.","DOI":"10.3390\/sym15030748"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1016\/j.mcm.2009.11.006","article-title":"Nabla discrete fractional calculus and nabla inequalities","volume":"51","author":"Anastassiou","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"823","DOI":"10.7153\/jmi-09-68","article-title":"Some new discrete fractional inequalities and their applications in fractional difference equations","volume":"9","author":"Zheng","year":"2015","journal-title":"J. Math. 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Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"830","DOI":"10.3390\/math7090830","article-title":"On fractional operators and their classifications","volume":"7","author":"Baleanu","year":"2019","journal-title":"Mathematics"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"149","DOI":"10.3390\/math7020149","article-title":"Desiderata for fractional derivatives and integrals","volume":"7","author":"Hilfer","year":"2019","journal-title":"Mathematics"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4907","DOI":"10.1090\/proc\/15133","article-title":"A reverse Minkowski-type inequality","volume":"148","author":"Hug","year":"2020","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"407","DOI":"10.46793\/KgJMat2203.407B","article-title":"On the reverse Minkowski\u2019s integral inequality","volume":"46","author":"Benaissa","year":"2022","journal-title":"Kragujevac J. Math."},{"key":"ref_25","first-page":"319","article-title":"A further generalization of the reverse Minkowski-type inequality via H\u00f6lder and Jensen inequalities","volume":"15","author":"Benaissa","year":"2022","journal-title":"J. Sib. Fed. Univ. Math. Phys."},{"key":"ref_26","first-page":"9","article-title":"Reverse H\u00f6lder and Minkowski-type inequalities for n functions","volume":"19","author":"Krasopoulos","year":"2022","journal-title":"Austral. J. Math. Anal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Lovric, M. (2011). International Encyclopedia of Statistical Science, Springer.","DOI":"10.1007\/978-3-642-04898-2"},{"key":"ref_28","first-page":"93","article-title":"Sur les expressions approximatives des integrales definies par les autres prises entre les m\u00eames limites","volume":"2","author":"Chebyshev","year":"1882","journal-title":"Proc. Math. Soc. Charkov"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"551","DOI":"10.7153\/jmi-07-51","article-title":"A variant of Chebyshev inequality with applications","volume":"7","author":"Liu","year":"2013","journal-title":"J. Math. Inequal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1609","DOI":"10.1007\/s13370-014-0312-5","article-title":"Some new Chebyshev type inequalities for functions whose derivatives belongs to Lp spaces","volume":"26","author":"Set","year":"2015","journal-title":"Afr. Mat."},{"key":"ref_31","first-page":"29","article-title":"A note on Chebyshev-Gr\u00fcss type inequalities for differential functions","volume":"22","author":"Pachpatte","year":"2006","journal-title":"Tamsui Oxf. J. Math. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"110860","DOI":"10.1016\/j.chaos.2021.110860","article-title":"Chebyshev type inequalities by using generalized proportional Hadamard fractional integrals via Polya\u2013Szeg\u00f6 inequality with applications","volume":"146","author":"Set","year":"2021","journal-title":"Chaos Solit. Fractals"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1090\/S0273-0979-02-00941-2","article-title":"The Brunn-Minkowski inequality","volume":"39","author":"Gardner","year":"2002","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.difgeo.2015.03.002","article-title":"Reverse Lp-dual Minkowski\u2019s inequality","volume":"40","author":"Zhao","year":"2015","journal-title":"Differ. Geom. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1007\/s00039-007-0619-6","article-title":"The Brascamp-Lieb inequalities: Finiteness, structure and extremals","volume":"17","author":"Bennett","year":"2008","journal-title":"Geom. Funct. Anal."},{"key":"ref_36","first-page":"532","article-title":"On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures","volume":"171","author":"Bobkov","year":"2000","journal-title":"J. Funct. Anal."},{"key":"ref_37","first-page":"232","article-title":"Reverse Minkowski inequality for risk measures","volume":"174","author":"Jiao","year":"2017","journal-title":"J. Optim. Theory Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"51","DOI":"10.15352\/afa\/1399900993","article-title":"On Minkowski and Hermite\u2013Hadamard integral inequalities via fractional integral","volume":"1","author":"Dahmani","year":"2010","journal-title":"Ann. Funct. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"148102","DOI":"10.1155\/2010\/148102","article-title":"On the Hermite\u2013Hadamard inequality and other integral inequalities involving two functions","volume":"2010","author":"Set","year":"2010","journal-title":"J. Inequal. Appl."},{"key":"ref_40","first-page":"165","article-title":"New fractional inequalities via Hadamard fractional integral","volume":"5","author":"Chinchane","year":"2013","journal-title":"Int. J. Funct. Anal. Oper. Theory Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"131","DOI":"10.3934\/Math.2018.1.131","article-title":"The Minkowski\u2019s inequality by means of a generalized fractional integral","volume":"3","year":"2018","journal-title":"AIMS Ser. Appl. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1186\/s13662-019-2229-7","article-title":"The Minkowski inequalities via generalized proportional fractional integral operators","volume":"2019","author":"Rahman","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1112\/plms\/s2-27.1.389","article-title":"The asymptotic expansion of generalized hypergeometric functions","volume":"27","author":"Fox","year":"1928","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1112\/jlms\/s1-10.40.286","article-title":"The asymptotic expansion of the generalized hypergeometric function","volume":"10","author":"Wright","year":"1935","journal-title":"J. Lond. Math. Soc."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2294","DOI":"10.3390\/sym13122294","article-title":"A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics","volume":"13","author":"Srivastava","year":"2021","journal-title":"Symmetry"},{"key":"ref_46","first-page":"423","article-title":"The asymptotic expansion of integral functions defined by Taylor series","volume":"238","author":"Wright","year":"1940","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Kashuri, A., Mohammed, P.O., and Baleanu, D. (2021). Fractional integral inequalities for exponentially nonconvex functions and their applications. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030080"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"135","DOI":"10.55579\/jaec.202153.340","article-title":"An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions","volume":"5","author":"Srivastava","year":"2021","journal-title":"J. Adv. Engrg. Comput."},{"key":"ref_49","first-page":"1501","article-title":"Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations","volume":"22","author":"Srivastava","year":"2021","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_50","first-page":"193","article-title":"On the generalized Hermite\u2013Hadamard inequalities","volume":"47","author":"Sarikaya","year":"2020","journal-title":"Ann. Univ. Craiova Math. Comput. Sci. Ser."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Kashuri, A., Mohammed, P.O., and Nonlaopon, K. (2021). Certain inequalities pertaining to some new generalized fractional integral operators. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040160"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"131","DOI":"10.3390\/fractalfract6030131","article-title":"Reverse Minkowski inequalities pertaining to new weighted generalized fractional integral operators","volume":"6","author":"Liko","year":"2022","journal-title":"Fractal Fract."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications.","DOI":"10.1063\/1.3047921"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1006\/jmaa.1997.5226","article-title":"Bounds for certain harmonic sums","volume":"206","author":"English","year":"1997","journal-title":"J. Math. Anal. Appl."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"264652","DOI":"10.1155\/2014\/264652","article-title":"New inequalities for gamma and digamma functions","volume":"2014","author":"Farhangdoost","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.7153\/jmi-2022-16-70","article-title":"Reverse form of the Minkowski inequalities with applications","volume":"16","author":"Pan","year":"2022","journal-title":"J. Math. Inequal."},{"key":"ref_57","first-page":"97","article-title":"On some reverses of Minkowski\u2019s, H\u00f6lder\u2019s and Hardy\u2019s type inequalities using \u03c8-fractional integral operators","volume":"18","author":"Yewale","year":"2022","journal-title":"South East Asian J. Math. Math. Sci."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1515\/ms-2017-0395","article-title":"On reverse H\u00f6lder and Minkowski inequalities","volume":"70","author":"Zhao","year":"2020","journal-title":"Math. 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