Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization

Abstract

This article is mainly concerned to link the Hermite-Hadamard and the Jensen-Mercer inequalities by using majorization theory and fractional calculus. We derive the Hermite-Hadamard-Jensen-Mercer-type inequalities in conticrete form, which serve as both discrete and continuous inequalities at the same time, for majorized tuples in the framework of the famous Atangana-Baleanu fractional operators. Also, the main inequalities encompass the previously established inequalities as special cases. Using generalized Mercer’s inequality, we also investigate the weighted forms of our major inequalities for certain monotonic tuples. Furthermore, the derivation of new integral identities enables us to construct bounds for the discrepancy of the terms concerning the main results. These bounds are constructed by incorporating the convexity of ∣f′∣| f^{\prime} | and ∣f′∣q(q>1){| f^{\prime} | }^{q}\hspace{0.33em}\left(q\gt 1) and making use of power mean and Hölder inequalities along with the established identities.

Keywords

• convex function
• majorization
• atangana-baleanu fractional operators
• 26a51
• 26d15
• 68p30