Integral inequalities involving fractional integral operators for certain differentiable types of convex function classes

In this thesis, new integral inequalities involving fractional integral operators have been obtained for certain differentiable classes of convex functions. The concept of convex functions plays a central role in many areas of analysis, as well as having significant applications in inequality theory. In this context, specific types of convexity—such as exponential convexity, trigonometric convexity, $s$-convexity, and $h$-convexity—have been considered, and generalized forms of Hermite-Hadamard type inequalities have been derived for these function classes. In the first chapter, the historical development of the concept of convexity is presented. Starting from classical convexity, studies in the literature on various types of convex functions have been evaluated, and the role of these functions in integral inequalities has been discussed. This chapter is important for introducing the foundational research that guides the topic of the thesis. The second chapter, titled Theoretical Foundations, includes definitions and theorems frequently used in the proof processes of the study. Relevant concepts from functional analysis are presented in detail to provide the theoretical groundwork for the arguments developed in subsequent sections. The third chapter, Materials and Methods, presents specific lemmas, theorems, and technical methods previously established in the literature, which guide the research findings. This chapter systematically explains the analytical techniques used in the study and the mathematical structures applied to generalize Hermite-Hadamard type inequalities. In the fourth chapter, titled Research Findings, original contributions of the thesis are introduced in the form of new theorems. Various integral inequalities have been obtained for exponential, trigonometric, $h$-convex, and $s$-convex function classes. Hölder, Young, and Power Mean inequalities have been applied to these results to determine their bounds. Additionally, results obtained for specific special functions and parameter selections are presented. The fifth and final chapter, Discussion and Conclusion, emphasizes the original aspects and contributions of the study. The limitations of the work are discussed, and suggestions for future research directions are provided. In conclusion, this thesis presents new and comprehensive integral inequalities involving fractional integral operators for differentiable functions with various convexity properties, offering significant contributions to both inequality theory and the literature on convex analysis.