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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. |