الفهرس 
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Abstract
Summary
The study of convex functions in recent years has received great at- tention from many researchers for its importance in various fields of pure and applied sciences such as economics, mechanics, and optimiza- tion, so the classical concepts of convex functions have been generalized in different directions. This thesis is concerned with two of the classes of generalized convex functions in the sense of Beckenbach, which are called sub E-convex functions and sub L-convex functions. We study the basic characteristics of these classes and generalize some of the integral properties and inequalities such as Hermite - Hadamard’s in- equality. The finding of the conjugate for these functions and the relationships between these functions and their conjugate.
This thesis aims to
1- Discuss two classes of the generalized convex functions in the sense of Beckenbach.
2- Study the main characterizations of sub E-convex functions.
3- Present the notion of conjugate sub E-convex function and derive certain properties of this function.
4- Extend some properties and integral inequalities (such as: Hermite- Hadamard) which are known for ordinary convex functions.
5- Propose a concept of sub L-convex functions and its supporting functions.
6- Study the main characterizations of sub L-convex functions.
7- Show the notion of conjugate sub L-convex function and obtain cer- tain properties of this function.
8- Introduce relationships between sub L-convex and sub E-convex
functions.
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. Also, we show the clas- sical convex functions and classes of generalized convex functions. In additions, the conjugate convex functions used to solve aircraft perfor- mance problems, such as maximum endurance and maximum range.
Chapter 2
In this chapter, we present some characteristics of sub E-convex functions. Also, we get the relationships between the class of sub E-convex functions and the class of positive increasing functions. Fur- thermore, we derive the integral inequalities of the Hadamard type for the multiplication of sub E-convex functions.
The results of this chapter are:
• under submission for publication.
Chapter 3
The aim of this chapter is to prove the supremum of sub E-convex functions is sub E-convex function. Also, we introduce the definition of the conjugate of sub E-convex functions, which plays an important role in linking the concept of duality among sub E-convex functions.
In addition, we give some relationships between sub E-convex func- tions and its conjugate. Furthermore, some properties for this class are established.
The results of this chapter are:
• Published in Advances in Mathematics: Scientific Journal, Vol. 10, No. 6, 2021. [15]
• Presented in the 3rd International Conference for Mathematics and Its Applications, 2020.
Chapter 4
This chapter aims to present a study that is mainly concerned with one class of generalized convex functions in the sense of Beck- enbach (sub L-convex functions). The existence of the support curves is presented for this class, which leads to its generalized convexity. In addition, an extremum property of these functions is given. Further- more, Hadamard’s inequality for this class is obtained.
The results of this chapter are:
• Published in International Journal of Applied Mathematics, Vol. 35, No. 4 (2022). [6]
Chapter 5
Finally, the goal of this chapter is to present the concept of the conjugate of sub L-convex functions. The relationships between the sub L-convex functions and their conjugate are considered, which lead to an important role in the notion of duality and connection between many of sub L-convex functions. In addition, we establish certain properties for this family. Also, we deduce the relationships between sub L-convex functions, sub E-convex functions and ordinary convex functions.
The results of this chapter are:
• under submission for publication.