الفهرس 
PreliminariesOnly 14 pages are availabe for public view
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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,
P´olya, Stefensen, 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 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 present a short survey of some needed
definitions, 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 definition of conju gate trigonometrically -convex functions by using Young’s inequality
which plays an important role in linking the concept of duality be-
tween trigonometrically -convex functions, rather the definition given
by Fenchel. Furthermore, we show that the integration of any increas-
ing 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.
Chapter 4
In this chapter, we derive several P´olya, Stefensen 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 mul-
tiplication 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 in-
equalities 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 trigono-
metrically convex functions. There are many applications of trigono-
metrically convex functions for examples in hydrofoils, geometry and extremum property. We show some applications as design of cavitation-
free hydrofoils by a given pressure envelope.
Ahydrofoil is simply a lifting surface, or foil, that operates in wa- ter. 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
decreases drag and allows greater speeds. The hydrofoils used exten-
sively 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 modifications depends on the
maximum velocity that is trigonometrically convex function.
In [24], a problem in geometry solved by using properties of trigono-
metrically convex function.
There exist another application in [2], which introduced that the
integration of diference between trigonometrically convex function and
its supporting function has a minimum value at middle of the interval.