{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T13:53:58Z","timestamp":1780408438055,"version":"3.54.1"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,8]],"date-time":"2020-04-08T00:00:00Z","timestamp":1586304000000},"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 inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of \u03bb -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite\u2013Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann\u2013Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.<\/jats:p>","DOI":"10.3390\/sym12040595","type":"journal-article","created":{"date-parts":[[2020,4,9]],"date-time":"2020-04-09T14:42:03Z","timestamp":1586443323000},"page":"595","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":99,"title":["On the Generalized Hermite\u2013Hadamard Inequalities via the Tempered 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, Kurdistan Region, Sulaimani 46001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mehmet Zeki","family":"Sarikaya","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, D\u00fczce University, D\u00fczce 81620, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 06530, Turkey"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan"},{"name":"Institute of Space Sciences, P.O. Box, MG-23, R 76900 Magurele-Bucharest, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,8]]},"reference":[{"key":"ref_1","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_2","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. 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