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In the present study, several new generalized integral inequalities of the Hadamard and Simpson-type are obtained. The results were obtained for functions whose first and third derivatives are either convex or satisfy the Lipschitz condition or the conditions of the Lagrange theorem. In a particular case, these results not only confirm but also improve some upper bounds, well known in the literature for the Simpson and Hermite-Hadamard-type inequalities.

In this paper, we establish some refinements and reverses of the celebrated F´ejer’s inequalities for the general case of functions defined on linear spaces. The obtained bounds are in terms of the Gˆateaux lateral derivatives. Some applications for norms and semi-inner products in normed linear spaces are also provided.

A characterization of h-convex function via Hermite-Hadamard inequality related to the h-convex functions is investigated. In fact it is determined that under what conditions a function is h-convex, if it satisfies the h-convex version of Hermite-Hadamard inequality.