The proportional-Caputo hybrid operator and some inequalities obtained with this operator

In this thesis, The proportional-Caputo hybrid operator obtained as a linear combination of the Caputo fractional derivative and the Riemann-Liouville integral is examined. The thesis consists of five main sections. In the first section, the historical development and significance of topics such as inequality, convexity, fractional derivatives, and integrals are emphasized. The second section provides the fundamental definitions and theorems related to these topics. The third section presents some existing fractional derivative and integral operators along with certain theorems found in the literature. In the fourth section, new inequalities obtained using The proportional-Caputo hybrid operator are presented. In the process of obtaining the findings, inequalities are initially established for different types of convex functions using the definition of The proportional-Caputo hybrid operator. Then, new inequalities are generated using an integral identity. Finally, some new results are obtained for bounded functions. In the fifth chapter, the conclusions and recommendations are shared.