The main motivation of the paper is to provide new integral inequalities for different kinds of convex functions by using a fractional integral operator with a non-singular kernel. The findings involve several new integral inequalities for quasi-convex functions and \((h,m)\)-convex functions. We have used the algebraic properties of Caputo-Fabrizio fractional operator, definitions of convex functions, and elementary analysis methods for the

In this study, the modification of the concept of exponentially convex function, which is a general version of convex functions, given on the coordinates, is recalled. With the help of an integral identity which includes the Riemann-Liouville (RL) fractional integral operator, new Hadamard-type inequalities are proved for exponentially convex functions on the coordinates. Many special cases of the results are discussed.

The main purpose of this study is to define geometrically exponentially convex functions, which are a more general version, by expanding geometrically convex functions and to create the relevant lemmas. Some properties of geometrically exponentially convex functions are proven using definitions and lemmas. While obtaining the main findings, in addition to basic analysis information, Young and Hölder inequalities, well known in the literature, were also used for the powers of some functions. In the new theorems obtained, some special results were obtained for α=0.

This study establishes new fractional integral inequalities associated with a certain class of differentiable functions. A composite fractional integral identity is derived by applying the fractional integral identity from Lemma 1 to the subintervals of a uniform partition. This identity serves as the main analytical tool in the subsequent analysis. By systematically combining this identity with Jensen-Mercer, Hölder, power-mean, and Young-type inequalities, several refined Hermite-Hadamard (HH)-type estimates are established. The obtained results extend and unify several existing inequalities reported in the literature while yielding a variety of particular cases through appropriate choices of the parameters.

In this study, new integral inequalities related to the Caputo-Fabrizio fractional integral operator are obtained using the definitions of convex functions and α-star s- convex function classes. In the results obtained with α-star s-convex functions, , α^s=1 is taken to obtain special results that have already been added to the literature. The well-known Hölder and Young inequalities for convex functions and α-star s-convex functions are used. In the proof of the integral inequalities obtained, the definition of function classes and basic analysis knowledge are utilized.