{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T14:09:11Z","timestamp":1774015751574,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,12]],"date-time":"2020-04-12T00:00:00Z","timestamp":1586649600000},"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>Integral inequalities play a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods. Thus, the present days need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition. There is a strong relationship between convexity and symmetry. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the past few years. In this article, we firstly point out the known Hermite\u2013Hadamard (HH) type inequalities which are related to our main findings. In view of these, we obtain a new inequality of Hermite\u2013Hadamard type for Riemann\u2013Liouville fractional integrals. In addition, we establish a few inequalities of Hermite\u2013Hadamard type for the Riemann integrals and Riemann\u2013Liouville fractional integrals. Finally, three examples are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.<\/jats:p>","DOI":"10.3390\/sym12040610","type":"journal-article","created":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T03:10:01Z","timestamp":1586833801000},"page":"610","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":72,"title":["A New Version of the Hermite\u2013Hadamard Inequality for Riemann\u2013Liouville Fractional Integrals"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani Sulaimani 46001, Kurdistan Region, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9793-8278","authenticated-orcid":false,"given":"Iver","family":"Brevik","sequence":"additional","affiliation":[{"name":"Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,12]]},"reference":[{"key":"ref_1","unstructured":"Lakshmikantham, V., and Leela, S. (1969). Differential and Integral Inequalities: Theory and Applications: Volume I: Ordinary Differential Equations, Academic Press."},{"key":"ref_2","unstructured":"Walter, W. (1964). Differential and Integral Inequalities (Vol. 55), Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1186\/s13660-016-1067-3","article-title":"Some generalized Riemann-Liouville k-fractional integral inequalities","volume":"2016","author":"Agarwal","year":"2016","journal-title":"J. Inequal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.mcm.2007.09.017","article-title":"Opial type inequalities involving Riemann\u2013Liouville fractional derivatives of two functions with applications","volume":"48","author":"Anastassiou","year":"2008","journal-title":"Math. Comput. Model."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1016\/j.camwa.2009.05.012","article-title":"Fractional integral inequalities and applications","volume":"59","author":"Denton","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"21","DOI":"10.2478\/v10294-012-0011-5","article-title":"Fractional integral inequalities for differential 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_7","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science Inc.. North-Holland Mathematics Studies, 204."},{"key":"ref_8","unstructured":"Miller, S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons."},{"key":"ref_9","first-page":"171","article-title":"Etude sur les proprietes des fonctions entieres et en particulier d\u2019une fonction considree par, Riemann","volume":"58","author":"Hadamard","year":"1893","journal-title":"J. Math. Pures. Appl."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"125","DOI":"10.12691\/tjant-6-4-5","article-title":"On New Trapezoid Type Inequalities for h-convex Functions via Generalized Fractional Integral","volume":"6","author":"Mohammed","year":"2018","journal-title":"Turkish J. Anal. Number Theory"},{"key":"ref_12","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":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.18514\/MMN.2017.1197","article-title":"On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals","volume":"17","author":"Sarikaya","year":"2017","journal-title":"Miskolc Math. Notes"},{"key":"ref_14","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_15","doi-asserted-by":"crossref","first-page":"112740","DOI":"10.1016\/j.cam.2020.112740","article-title":"On generalized fractional integral inequalities for twice differentiable convex functions","volume":"372","author":"Mohammed","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1186\/s13662-020-2541-2","article-title":"Modification of certain fractional integral inequalities for convex functions","volume":"2020","author":"Mohammed","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Fernandez, A., and Mohammed, P.O. (2020). Hermite-Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels. Math. Meth. Appl. Sci., 1\u201318.","DOI":"10.1002\/mma.6188"},{"key":"ref_18","first-page":"33","article-title":"On some Ostrowski type inequalities","volume":"18","author":"Gavrea","year":"2010","journal-title":"Gen. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1186\/s13660-019-2079-6","article-title":"Generalized fractional integral inequalities of Hermite\u2013Hadamard type for (\u03b1,m)-convex functions","volume":"2019","author":"Qi","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"287947","DOI":"10.1155\/2008\/287947","article-title":"Hermite-Hadamard Inequality on Time Scales","volume":"2008","author":"Dinu","year":"2008","journal-title":"J. Inequal. Appl."},{"key":"ref_21","first-page":"135","article-title":"Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions","volume":"4","author":"Mohammed","year":"2016","journal-title":"Turkish J. Anal. Number Theory"},{"key":"ref_22","first-page":"199","article-title":"Inequalities of (k, s), (k, h)-Type For Riemann-Liouville Fractional Integrals","volume":"17","author":"Mohammed","year":"2017","journal-title":"Appl. Math. E-Notes"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1016\/j.jksus.2017.07.011","article-title":"Some new Hermite-Hadamard type inequalities for MT-convex functions on differentiable coordinates","volume":"30","author":"Mohammed","year":"2018","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O. (2019). Hermite-Hadamard inequalities for Riemann-Liouville fractional integrals of a convex function with respect to a monotone function. Math. Meth. Appl. Sci., 1\u201311.","DOI":"10.1186\/s13660-019-1982-1"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1186\/s13660-018-1950-1","article-title":"Hermite-Hadamard type inequalities for \u03dd-convex function involving fractional integrals","volume":"2018","author":"Mohammed","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_26","first-page":"93","article-title":"On generalization integral inequalities for fractional integrals","volume":"25","author":"Sarikaya","year":"2014","journal-title":"Nihonkai Math. J."},{"key":"ref_27","first-page":"227","article-title":"New fractional inequalities of Ostrowski-Gr\u00fcss type","volume":"LXIX","author":"Sarikaya","year":"2014","journal-title":"Le Mat."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"168008","DOI":"10.1016\/j.aop.2019.168008","article-title":"Self-stress on a dielectric ball and Casimir-Polder forces","volume":"412","author":"Milton","year":"2020","journal-title":"Ann. Phys. (N. Y.)"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"085010","DOI":"10.1103\/PhysRevD.96.085010","article-title":"Electromagnetic delta-function sphere","volume":"96","author":"Parashar","year":"2017","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"085006","DOI":"10.1103\/PhysRevD.60.085006","article-title":"Casimir surface force on a dilute dielectric ball","volume":"60","author":"Brevik","year":"1999","journal-title":"Phys. Rev. D"},{"key":"ref_31","unstructured":"Watson, G.N. (1944). A Treatise on the Theory of Bessel Functions, Cambridge University Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/610\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:08:51Z","timestamp":1760364531000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/610"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,12]]},"references-count":31,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040610"],"URL":"https:\/\/doi.org\/10.3390\/sym12040610","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,12]]}}}