The main aim of this study is to reveal new generalized-Ostrowski-type inequalities using Katugampola fractional integral operator which generalizes Riemann-Liouville and Hadamard fractional integral operators into a single form. For this purpose, at first, a new fractional integral identity is generated by the researchers. Then, by using this identity, some inequalities for the class of functions whose certain powers of absolute values of derivatives are $p-$convex are derived. Some applications to special means for positive real numbers are also given. It is observed that the obtained inequalities are generalizations of some well known results.

Fractional analysis has evolved considerably over the last decades and has become popular in many technical and scientific fields. Many integral operators which ables us to integrate from fractional orders has been generated. Each of them provides different properties such as semigroup property, singularity problems etc. In this paper, firstly, we obtained a new kernel, then some new integral inequalities which are valid for integrals of fractional orders by using Riemann-Liouville fractional integral. To do this, we used some well-known inequalities such as Hölder's inequality or power mean inequality. Our results generalize some inequalities exist in the literature.

In this paper, we establish several new inequalities for twice differentiable mappings that are connected with the celebrated Hermite-Hadamard integral inequality.

In this paper, we introduce the notion of ( g , φ h ) -convex dominated function and present some properties of them. Finally, we present a version of Hermite-Hadamard-type inequalities for ( g , φ h ) -convex dominated functions. Our results generalize the Hermite-Hadamard-type inequalities in Dragomir et al. (Tamsui Oxford Univ. J. Math. Sci. 18(2):161-173, 2002), Kavurmacı et al. (New Definitions and Theorems via Different Kinds of Convex Dominated Functions, 2012) and Özdemir et al. (Two new different kinds of convex dominated functions and inequalities via Hermite-Hadamard type, 2012). MSC:26D15, 26D10, 05C38.

This study is built on the relationship between inequality theory and fractional analysis. Thanks to the new fractional operators and based on the proportional Caputo-hybrid operators, integral inequalities containing new approaches are obtained for differentiable convex functions. In the findings section, firstly, an integral identity is obtained and various integral inequalities are obtained based on this identity. The peculiarity of the results is that a hybrid operator has been used in inequality theory, which includes the derivative and integral operators together.

The primary objective of this study is to handle new generalized Hermite–Hadamard type inequalities with the help of the Katugampola fractional integral operator, which generalizes the Hadamard and Riemann–Liouville fractional integral operators into one system. In order to do this, a new fractional integral identity is obtained. Then, by using this identity, some inequalities for the class of functions whose derivatives in absolute values at certain powers are ρ-convex are derived. It is observed that the obtained inequalities are generalizations of some results in the literature.