A Study on Some Classes of Generalized Convex Functions

Abstract

The aim of this thesis is to 1- Discuss some classes of the generalized convex functions in the sense of Beckenbach. 2- Study the main characterizations of sub E-convex functions. 3- Extend some properties and integral inequalities (such as: Hermite- Hadamard, Hermite-Hadamard-Feje´r, Ostrowski and Trapezoid, ...) which are known for ordinary convex functions. 4- Show that some results introduced by Hu¨seyin Budak [11], in (2019), are incorrect.

The thesis consists of five chapters: Chapter 1 This chapter is an introductory chapter. It contains definitions and basic concepts that are used throughout this thesis. It is regarded as a short survey of the basic needed material. Chapter 2 The goal of this chapter is to obtain some new inequalities of Hermite-Hadamard and Hermite-Hadamard-Feje´r type inequalities via fractional integrals for trigonometric ρ-convex functions. Furthermore, we use the Riemann-Liouville fractional integral to present recent re- sults on fractional integral inequalities for trigonometric ρ-convex func- tions. Also, we show that some results introduced by Hu¨seyin Budak [11], in (2019), are incorrect. Moreover, a counter example is given to confirm our claim. The results of this chapter are:

• under submission for publication. Chapter 3 The purpose of this chapter is to get upper and lower estimates for product of two hyperbolic p-convex functions, which is analogous to Hermite-Hadamard type inequalities for product of two hyperbolic p-convex functions. The results of this chapter are: • Accepted for publication in Italian Journal of Pure and Applied Mathematics on August 3rd, 2021. • Presented in the 3rd International Conference for Mathematics and Its Applications, 2020. Chapter 4 The aim of this chapter is to study the standard functional op- erations of sub E-convex functions. Furthermore, we introduce a class BE[a, b] of functions representable as a difference of two sub E-convex functions. The results of this chapter are: Published in RGMIA Research Report Collection, Vol. 25, No. 2, 2022. Chapter 5 Finally, in this chapter, we show that the power function of sub E-convex function f n(x) is sub E-convex function. Furthermore, we establish some new integral inequalities of Hadamard type involving sub E-convex functions and sub E-concave functions. The results of this chapter are: Published in Advances in Mathematics: Scientific Journal, Vol. 9, No. 3, 2020.