Application of fractional derivatives and integrals to some inequalities

Fractional derivatives and integrals are a very popular topic. Although the concept of fractional derivative and integral is popular, it is known that the first studies on this subject were in the 1600s. In this study, it is explained how the concept of fractional derivative and integral emerged. Throughout history, scientists working on fractional derivatives and integrals have been introduced. Scientists have expressed different points of view on this issue. Then, important definitions and theorems are discussed in order to better understand fractional derivatives and integrals. Again, the Riemann-Liouville fractional derivative and integral, Grünwald Letnikov fractional derivative and Caputo fractional derivative, which have the most important place among fractional derivatives and integrals, are defined, their properties are given, fractional derivatives of some special functions are examined and examples of them are given. In addition, their fractional derivative change graph was drawn. Finally, some inequalities for convex functions are discussed by means of fractional integrals. It has been proven that these inequalities are provided and practices related to them have been made.