The aim of inequalities is to develop tools for analyzing the problems in pure and applied mathematics. Our primary objective in this research is to introduce some new type inequalities connected with Caputo Fractional derivative. In the light of the operator, we extend and generalize some important inequalities to this Fractional calculus dealing with synchronous functions.

In this paper, we , rst establish two new identities for differentiable function involving generalized Fractional integrals. Then, by utilizing these equalities, we obtain some midpoint type inequalities involving generalized Fractional integrals for mappings whose derivatives in absolute values are convex. We also give several results as special cases of our main results.

Fejer Hadamard inequality is generalization of Hadamard inequality. In this paper we prove certain Fejer Hadamard inequalities for k-Fractional integrals. We deduce Fejer Hadamard-type inequalities for Riemann-Liouville Fractional integrals. Also as special case Hadamard inequalities fork-Fractional as well as Fractional integrals are given.

In the literature, several papers are devoted to inequalities of Simpson-type in the case of differentiable convex functions and Fractional versions. Moreover, some papers are focused on inequalities of Simpson-type for twice differentiable convex functions. In this research article, we obtain an identity for twice differentiable convex functions. Then, we prove several Fractional inequalities of Simpson-type for convex functions.

In this article we present applications of Hardy-type and refined Hardy-type inequalities for a generalized Fractional integral operator involving the Mittag-Leffler function in its kernel and for the Hilfer Fractional derivative using convex and monotone convex functions.