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In this study, a new definition of the fractional integral operator is first proposed, which generalizes some well-known fractional integral operators. Then, by using this newly generalized fractional integral operator, we proved several new Hermite-Hadamard-type inequalities for convex functions. Finally, we provided some corollaries to show that the current results extend and enrich the previous results in the literature. [ABSTRACT FROM AUTHOR]

In this investigation, we unfold the Jensen–Mercer ( J − M ) inequality for convex stochastic processes via a new fractional integral operator. The incorporation of convex stochastic processes, the J − M inequality and a fractional integral operator having an exponential kernel brings a new direction to the theory of inequalities. With this in mind, estimations of Hermite–Hadamard–Mercer ( H − H − M )-type fractional inequalities involving convex stochastic processes are presented. In the context of the new fractional integral operator, we also investigate a novel identity for differentiable mappings. Then, a new related H − H − M -type inequality is presented using this identity as an auxiliary result. Applications to special means and matrices are also presented. These findings are particularly appealing from the perspective of optimization, as they provide a larger context to analyze optimization and mathematical programming problems. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the endpoint boundedness for the commutators of singular integral operators with BMO functions and the associated maximal operators on weighted generalized Morrey spaces. We also get similar results for the commutators of fractional integral operators with BMO functions and the associated maximal operators. [ABSTRACT FROM AUTHOR]

In this article, a new general integral identity involving generalized fractional integral operators is established. With the help of this identity new Hermite-Hadamard type inequalities are obtained for functions whose absolute values of derivatives are convex. As a consequence, the main results of this paper generalize the existing Hermite-Hadamard type inequalities involving the Riemann-Liouville fractional integral. [ABSTRACT FROM AUTHOR]

We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the weighted estimates of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The main result generalizes the Sawano, Sugano, and Tanaka result to a weighted setting. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the mapping properties of the fractional integral operators on weighted modulation spaces. Based on this result, we also study the boundedness of the bilinear fractional integral operators on product Wiener amalgam spaces. Our results show that, besides modulation spaces, Wiener amalgam spaces are good substitutions for Lebesgue spaces. [ABSTRACT FROM AUTHOR]