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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.

In this paper, by introducing multiple parameters, we establish a discrete version of the Hardy–Littlewood–Polya inequality in the whole plane. For the obtained inequality, we give the equivalent statements of the best possible constant factor linked to the parameters and deal with the equivalent inequalities. Our main result provided a new generalization of Hardy–Littlewood–Polya inequality, and as a consequence, we show that some new inequalities of the Hardy–Littlewood–Polya type can be derived from the current results by taking the special values of parameters. [ABSTRACT FROM AUTHOR]

In this paper, a weighted generalized fractional integral operator based on the Mittag-Leffler function is established, and it exhibits symmetric characteristics concerning classical operators. We demonstrate the semigroup property as well as the boundedness of the operator in absolute continuous like spaces. In this work, some applications with graphical representation are also considered. Finally, we modify the weighted generalized Laplace transform and then applied it to the newly defined weighted fractional integral operator. The defined operator is an extension and generalization of classical Riemann–Liouville and Prabhakar integral operators. [ABSTRACT FROM AUTHOR]