الفهرس 
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Abstract
This thesis is devoted to
1. Discuss one of classes of the generalized convex functions in the
sense of Beckenbach which are known as trigonometrically -
convex functions.
2. Study the main characterization of trigonometrically -convex
functions.
3. Extend some properties and integral inequalities such as: Young,
Polya, Ste ensen, Hermite-Hadamard, Cauchy-Schwarz.
4. Introduce applications of trigonometrically convex functions.
The thesis consists of six chapters:
Chapter 1
This chapter is an introductory chapter. It contains de nitions
and basic concepts that are used throughout this thesis. It is regarded
as a short survey of the basic needed material.
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SUMMARY
Chapter 2
The goal of this chapter is to present a short survey of some needed
de nitions, basic concepts and results of these two important vital
topics: trigonometrically -convex functions and supporting functions.
Also, some integral inequalities for Hermite-Hadamard and for higher
powers of trigonometrically -convex functions are showed.
Chapter 3
The purpose of this chapter is to introduce a de nition of conjugate
trigonometrically -convex functions by using Young’s inequality
which plays an important role in linking the concept of duality between
trigonometrically -convex functions, rather the de nition given
by Fenchel. Furthermore, we show that the integration of any increasing
functions are trigonometrically -convex functions.
Some results of this chapter are:
Accepted in Italian Journal of Pure and Applied Mathematics,
on December 22, 2018.
Presented in the 2nd National Conference for Mathematics and
Applications, Cairo, Egypt, 2017.
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SUMMARY
Chapter 4
In this chapter, we derive several Polya, Ste ensen and Hermite-
Hadamared type integral inequalities for trigonometrically -convex
functions.
Some results of this chapter are:
Published in International Journal of Applied Mathematics, Vol. 31,
No 6 (2018), pp. 779-795.
Chapter 5
The aim of this chapter is to study some properties of the multiplication
of two trigonometrically -convex functions, and prove the
non negative convex function is trigonometrically -convex functions.
Furthermore, we establish several Cauchy-Schwarz’s type integral inequalities
for trigonometrically -convex functions. The results of this
chapter are under submission for puplication.
Chapter 6
The content of this chapter is to introduce applications of trigonometrically
convex functions. There are many applications of trigonometrically
convex functions for examples in hydrofoils, geometry and
extremum property. We show some applications as design of cavitationfree
hydrofoils by a given pressure envelope.
A hydrofoil is simply a lifting surface, or foil, that operates in water.
These are similar to aerofoils used in aeroplanes. As a hydrofoil
craft gains speed, the hydrofoils lift the boats hull out of the water. It
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SUMMARY
decreases drag and allows greater speeds. The hydrofoils used extensively
during the First World War by American. In [8], they describe
basic aspects of the theory of pressure which allows to modify a series
of hydrofoils designed by Eppler. This modi cations depends on the
maximum velocity that is trigonometrically convex function.
In [24], a problem in geometry solved by using properties of trigonometrically
convex function.
There exist another application in [2], which introduced that the
integration of di erence between trigonometrically convex function and
its supporting function has a minimum value at middle of the interval