Some new variants of fractional Hermite-Hadamard and Pachpatte-type integral inequalities involving Raina's and Mittag-Leffer functions with applications

Volume 40, Issue 3, pp 415--443 https://dx.doi.org/10.22436/jmcs.040.03.08

Publication Date: July 24, 2025 Submission Date: May 26, 2025 Revision Date: June 09, 2025 Accteptance Date: June 26, 2025

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Authors

M. Tariq - Mathematics Research Center, Near East University, Near East Boulevard, PC: 99138, Nicosia/Mersin 10, Turkey. - Department of Mathematics, Balochistan Residential College, Loralai, Balochistan, Pakistan. S. K. Ntouyas - Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece. J. Tariboon - Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand.

Abstract

Integral inequalities combined with convexity in the frame of fractional calculus is an interesting research topic. Mathematical inequalities and convex functions have become vital to the growth of many pure and applied fields of science. In this article, we demonstrate a few generalized fractional integral inequalities involving Raina's function that represent the Mittag-Leffler function. This article provides to an intriguing connection between convex functions, special functions, and fractional calculus. First, we present and investigate the concept of a generalized convex involving Raina's function in a polynomial context and discuss its algebraic properties. We establish the new mathematical approach of Hermite-Hadamard inequality and Pachpatte-type inequality involving the newly introduced definition via Caputo-Fabrizio fractional integral operator. Furthermore, to improve our results, we establish a new fractional lemma and utilizing this, provides some new fractional perspectives of the Hermite-Hadamard-type inequality with the aid of generalized m-convex function involving Raina's function. Applications of some of our presented results to special means are given as well. The study's conclusions provide fresh and noteworthy improvements over previous research, offering special perspectives and contributions to the area.

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Keywords

• Convex function
• Raina function
• Generalized convex involving Raina's function
• improved power-mean integral inequality
• Hölder-İşcan inequality
• Caputo-Fabrizio operator

MSC

•  26A51
•  26A33
•  26D07
•  26D10
•  26D15

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