الفهرس | Only 14 pages are availabe for public view |
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, Steensen, 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 denitions and basic concepts that are used throughout this thesis. It is regarded as a short survey of the basic needed material. 7 SUMMARY Chapter 2 The goal of this chapter is to present a short survey of some needed denitions, 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 denition 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 denition 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. 8 SUMMARY Chapter 4 In this chapter, we derive several Polya, Steensen 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 9 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 modications 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 dierence between trigonometrically convex function and its supporting function has a minimum value at middle of the interval |